Master Thesis
The persistent information paradox
Author: Supervisor: Nick Bultinck Dr. Karel Van Acoleyen
A thesis submitted in fulfilment of the requirements for the degree of Master of science in de fysica en sterrenkunde
FACULTEIT WETENSCHAPPEN Vakgroep Fysica en Sterrenkunde
June 2013
UGENT
Abstract
Faculteit Wetenschappen Vakgroep Fysica en Sterrenkunde
Master of science in de fysica en sterrenkunde
The persistent information paradox
by Nick Bultinck
This thesis is about black holes and their turbulent but very fruitful relation with quantum mechanics. First it is explained why black holes can be seen as truly thermodynamical objects. This is done by looking at their classical properties and by doing quantum field theory around them. Just as ordinary thermodynamical objects black holes emit thermal radiation. Conser- vation of energy will cause them to shrink and to eventually evaporate completely. Although calculations in the semiclassical approach seem to suggest that the evaporation of a black hole evolves pure states into mixed states, we argue why and how this process should nevertheless be made unitary. This will lead us to the long-lived and well-established phenomenological descrip- tion of unitary black holes called black hole complementarity. However, it is shown that a closer look at the postulates of black hole complementarity reveals a paradox. In particular, there is a conflict between unitarity, the equivalence principle and effective field theory. It seems that if one is reluctant to give up unitarity, a destructive firewall should be placed at the horizon. Jumping into a black hole may be much more dangerous than expected.
Key words General relativity, black holes, quantum field theory, information paradox, black hole complementarity, quantum gravity, firewall
Acknowledgements
I would like to thank my supervisor Dr. Karel Van Acoleyen for the freedom he gave me and for the support I could always ask. I felt great confidence from his side. Also, I am very grateful that I have been granted the opportunity to study this fascinating subject. It was fun working on this thesis.
En dan uiteraard
Mijn moeder en mijn vader, Julie,
bedankt voor alles. Jullie zijn een grote steun voor mij.
iv
Contents
Abstract ii
Acknowledgements iv
Preliminary x
1 Classical aspects of black holes1 1.1 Spacetime symmetries...... 2 1.2 Null surfaces...... 5 1.3 The Cauchy problem in General Relativity...... 6 1.3.1 Differential equations in curved spacetime...... 6 1.3.2 Global Hyperbolicity...... 7 1.4 The Horizon...... 8 1.4.1 Gravitational collapse...... 8 1.4.2 Outside observers and the end of time...... 12 1.4.3 Infalling observers and the equivalence principle...... 14 1.5 The maximal extension of Schwarzschild spacetime...... 16 1.6 Killing horizons and surface gravity...... 19 1.7 Penrose diagrams...... 23 1.7.1 Conformal compactification...... 23 1.7.2 Gravitational collapse spacetime...... 28 1.8 No hair conjecture...... 30 1.8.1 Kerr-Newman geometry and uniqueness theorems...... 30 1.8.2 Classical fields...... 34 1.8.3 The electric Meissner effect and equilibrium...... 36 1.9 Radial null geodesics in black hole spacetimes...... 36 1.9.1 The Schwarzschild black hole...... 37 1.9.2 The Kerr black hole...... 39 1.10 Energy extraction in Kerr spacetime...... 44 1.10.1 The ergosphere of a Kerr black hole...... 44 1.10.2 The Penrose process...... 45 1.10.3 Superradiance...... 47 1.11 Black hole mechanics...... 49 1.11.1 The area theorem...... 50 1.11.2 Zeroth law...... 52 1.11.3 First law...... 53
vi Contents vii
2 Quantum field theory in curved spacetime 57 2.1 The formulation of QFT in curved spacetime...... 58 2.1.1 The canonical approach...... 58 2.1.2 A cosmological model...... 63 2.1.2.1 The set-up...... 63 2.1.2.2 Particle creation...... 67 2.1.2.3 Conclusions...... 68 2.1.3 The loss of Poincar´esymmetry...... 69 2.1.3.1 The particle content of the Klein-Gordon field...... 70 2.1.3.2 The lack of spacetime symmetries...... 70 2.1.3.3 The algebraic approach...... 73 2.2 The Unruh effect...... 73 2.2.1 Rindler spacetime...... 74 2.2.2 Accelerating observers and the thermal bath...... 77 2.3 Particle creation by black holes...... 82 2.3.1 Original derivation of the Hawking radiation...... 83 2.3.1.1 The Schwarzschild black hole...... 86 2.3.1.2 The Kerr black hole...... 91 2.3.1.3 Final remarks...... 92 2.3.2 Alternative views on the Hawking radiation...... 94 2.3.2.1 Static observers and the Unruh effect...... 94 2.3.2.2 Heuristic arguments...... 95 2.3.3 Trans-Planckian physics in Hawking radiation...... 96 2.4 Angular momentum and gray body factors...... 97 2.5 The generalized second law...... 100 2.5.1 The lowering of matter in a static black hole...... 101 2.5.2 A more general argument...... 105 2.6 Euclidean path integral methods...... 107 2.6.1 Hawking temperature derivation...... 108 2.6.2 Black hole entropy derivation...... 109
3 The membrane paradigm 115 3.1 The stretched horizon...... 115 3.2 A conducting surface...... 118 3.3 Spreading of a charge...... 121
4 Entanglement and information 125 4.1 Density matrices and entanglement...... 125 4.1.1 Ensembles...... 126 4.1.2 Quantum statistical mechanics...... 127 4.1.3 Reduced density matrix...... 130 4.2 Unruh density matrix...... 132 4.3 Generalized second law for quasistationary semiclassical black holes...... 134 4.4 The information paradox...... 137 4.4.1 A toy model...... 137 4.4.1.1 Nice slices...... 138 4.4.1.2 Particle creation revisited...... 139 Contents viii
4.4.1.3 Slicing the black hole geometry...... 140 4.4.1.4 From pure to mixed...... 143 4.4.1.5 Mixed states and information...... 146 4.4.2 The true physical situation...... 147 4.5 Implications of non-unitary evolution...... 148 4.5.1 The superscattering operator...... 148 4.5.2 A general evolution equation...... 149 4.5.3 A subclass of solutions...... 152 4.6 Possible ways to unitarity...... 154 4.6.1 Information in the Hawking radiation...... 154 4.6.1.1 Backreaction and small corrections...... 154 4.6.1.2 Quantum hair and fuzzballs...... 159 4.6.2 Stable remnants...... 159 4.6.3 Information release at the end...... 161 4.6.4 Baby universes...... 163 4.6.5 Other modifications of conventional theories...... 165 4.7 AdS/CFT and the information paradox...... 166 4.8 Euclidean gravity and unitarity...... 167 4.9 Thermodynamics of horizons...... 170 4.10 Horizon entanglement entropy...... 179 4.10.1 Entanglement entropy in flat spacetime...... 180 4.10.2 Entanglement entropy of Killing horizons...... 185
5 Black hole complementarity 189 5.1 A brick wall...... 189 5.2 Problems with information in the Hawking radiation...... 193 5.3 Average information in the Hawking radiation...... 195 5.4 The postulates...... 197 5.5 Thought experiments...... 201 5.5.1 Verification of the stretched horizon...... 201 5.5.2 Baryon number violation...... 202 5.5.3 Entangled spins...... 204 5.6 Old black holes as quantum mirrors...... 207 5.7 Fast scrambling...... 212 5.7.1 Scrambling in general quantum systems...... 212 5.7.2 Scrambling in black holes...... 215 5.7.3 The entangled spin experiment revisited...... 216 5.8 Complementarity in the semiclassical framework...... 217
6 The Firewall 223 6.1 The AMPS argument...... 223 6.1.1 The entropy argument...... 224 6.1.2 The projection argument...... 225 6.2 The thermal zone and mining...... 228 6.3 Why complementarity is not enough...... 229 6.4 Migrating singularity...... 230 6.5 Formation time of the firewall...... 233 Contents ix
6.5.1 Generic and scrambled...... 233 6.5.2 Fine grained and coarse grained...... 234 6.5.3 Special states and generic states...... 235 6.6 Non-local dynamics...... 238 6.7 The Harlow-Hayden conjecture...... 240 6.7.1 The decoding process...... 241 6.7.2 General unitary transformation with quantum gates...... 245 6.7.3 Decoding is slower than black hole dynamics...... 247 6.8 Next generation complementarity...... 249 6.8.1 The stretched horizon as a hologram...... 249 6.8.2 The transfer of (distillable) entanglement...... 251 6.8.3 Standard complementarity (A = RB ⊂ R)...... 254 6.8.4 Strong complementarity...... 255 6.9 Problems with A ⊂ R ...... 257 6.9.1 Measurements create a firewall...... 257 6.9.2 State dependence...... 259 6.9.3 Arbitrariness and energy considerations...... 260 6.10 The Hilbert space for an infalling observer...... 262 6.11 Static AdS black holes...... 264 6.12 Conclusion...... 265 6.13 * Personal view ...... 267 6.13.1 A firewall? ...... 268 6.13.2 Backreaction ...... 269 6.13.3 Freely falling vs. hovering ...... 270 6.13.4 Global vs. local ...... 273
A Frobenius’s theorem 277
B Surface gravity of a Kerr black hole 280
C The zeroth law 283
D The Hamiltonian formulation of general relativity 287
E Scrambled entanglement 292 E.1 Ordered ground states...... 292 E.2 Scrambled systems...... 293
Bibliography 297 Preliminary
General relativity. Quantum field theory. Two well established, beautiful and very succes- ful theories. Physicists have been trying to unify these theories since a very long time now. However, this appears to be a very difficult task because at the present time there is still no satisfactory quantum theory of gravity. This thesis deals with a more modest task: reconciling black holes with unitarity. But it nevertheless is the hope that a successful unitary picture of these gravitational objects can provide us with some clues of how to think about a unified theory of general relativity and quantum mechanics.
This thesis can be seen as being composed out of two parts. The first part consists of chapters 1, 2 and 3. The main purpose of chapter 1, which deals with some classical aspects of black holes, is to give the necessary background for chapter 2. In this second chapter quantum field theory in curved spacetime is examined in order to explain particle creation by black holes. In chapter 3 a very useful and intuitive mental picture of black holes will be presented which goes under the name of the membrane paradigm. This first part is the ’positive’ part. It shows how general relativity and quantum mechanics agree perfectly about the thermodynamic nature of black holes. Both theories merge beautifully in this area.
The second part consists of chapters 4, 5 and 6. These deal with the main purpose of this thesis, which is the information paradox and its ramifications. Actually, the first part could be skipped and one could simply start reading at chapter 4, just assuming some basic results of the first three chapters. However, it is my opinion that the first part provides a valuable context for the information paradox which helps to understand the depth of the matter being presented.
The second part is the ’negative’ part. While quantum mechanics and general relativity matched perfectly in the first part, their combination will give rise to paradoxes and conflicts of princi- ple in the second. This will force us to leave the conventional tracks and to go onto unknown slippery roads. While the first part was about finding the correct outcomes and interpretations of known formulas, the second part deals with the search for new formulas. Or not even that, it concentrates on the search for new principles that could possibly lead to new formulas. In this search thought experiments will prove themselves to be of great use.
Although there are many promising theories to explain quantum gravity like string theory
x Contents xi and loop quantum gravity, I will try to restrict to what we can learn from the experimentally confirmed theories general relativity and quantum field theory. This means I will also not ex- plicitely consider the fuzzball models, i.e. black hole models based on string theory. In some cases I will refer to the AdS/CFT correspondence, but only as to provide additional information to broaden the picture.
My aim was to make this thesis self-containing. If I have succeeded, then a graduate student like myself, who has had some introductory courses on general relativity, quantum mechanics and quantum field theory, should be able to follow everything without having to consult any of the references.
This thesis contains 50 years of physics, from the golden age of black hole physics in the 1960’s to the present day, with ingedients from a variety of fields.
I hope the reader may enjoy it as much as I did.
Nick Bultinck June 2013
PS: I would also like to apologize for the typos that escaped my eye and which are undoubtedly present in a work of this length. Chapter 1
Classical aspects of black holes
A luminous star, of the same density of the earth, and whose diameter should be two hundred and fifty times larger than that of the sun, would not, in consequence of its attraction, allow any of its rays to arrive at us. It is therefore possible that the largest luminous bodies in the universe may, through this cause, be invisible. - P.S. Laplace (1798)
Imagine a big, let’s say infinite, shallow lake. In this lake live some fish. Because of their biology these fish cannot swim faster than the speed of sound in the water. Now suppose that somewhere in this lake, there is a hole in the bottom, creating a drainhole through which water is flowing away. The water that is sucked away flows along some very sharp rocks, so everything that gets sucked in the drainhole encounters a very violent end of its existence. This situation is depicted in figure 1.1a.
Figure 1.1: Some fish in a lake.
As in every normal drainhole, the water flows faster and faster when approaching towards the center. At some point, the water starts flowing faster than the speed of sound. The line where this happens is depicted with a circle in figure 1.1b. Now suppose that a fish, called Alice, passes through this line. The moment when she passes this line, nothing unusual happens to Alice, she feels no other than before. But there is definitely a great consequence of this passage, because for an outside fish, say Bob, Alice has ceased to exist. Alice cannot swim back to inform Bob that she is fine because she cannot swim faster than the speed of sound, she is past the point of no return. It is also impossible to let Bob hear from her existence because the soundwaves are 1 Chapter 1. Classical aspects of black holes 2 also trapped behind this point of no return. So for Bob, living on the outside, Alice is gone. Of course, what is presented here is an analogy to a black hole (Remarkably, this analogy has a direct and practical application to the mechanism of Hawking radiation, see section 2.3.3). Black holes and their combination with quantum mechanics is the context in which the subject of this thesis is situated. In this first chapter the necessary classical aspects of black holes are presented. Black holes are objects pushing human imagination to its utter limits. But nevertheless, they may be quite less exotic as they would appear at first sight. Calculations show that a star of a mass that is one and a half times the mass of our sun has a fair chance of collapsing to a black hole at the end of its life [1]. And there are at least 109 more massive stars in our galaxy alone...
1.1 Spacetime symmetries
Consider a timelike curve C with endpoints A and B. The action for a free particle of mass m moving on C is Z B Z B S = −mc2 dτ = −mc ds, (1.1) A A where τ is the proper time on C. Since
p µ ν p µ ν ds = dx dx gµν = x˙ x˙ gµν dλ , (1.2)
µ dxµ where λ is an arbitrary affine parameter on C andx ˙ = dλ , one has
Z b p µ ν S[x(λ)] = −mc dλ x˙ x˙ gµν (c = 1) . (1.3) a The world lines of free particles or geodesics are found by demanding that
δS[x] = 0 . (1.4) δx(λ)
This implies Z b ∂L d ∂L σ 0 = δS = dλ σ − σ δx . (1.5) a ∂x dλ ∂x˙ Because this result should be independent of the variation one gets the Euler-Lagrange equations
d ∂L ∂L − = 0 . (1.6) dλ ∂x˙ σ ∂xσ
Which become explicitely
1 0 =x ¨κ + gρκ(∂ g + ∂ g − ∂ g )x ˙ µx˙ σ (1.7) 2 σ µρ µ σρ ρ µσ κ κ µ σ =x ¨ + Γµσx˙ x˙ (1.8)
And these are the geodesic equations. A solution xµ(λ) to these equations is called a geodesic. Chapter 1. Classical aspects of black holes 3
The tangent to a general curve C with equation xµ(λ) is given by tµ =x ˙ µ(λ). For a gen- eral vector field V µ(x(λ)) along C one has
ν µ ν µ ν µ ρ t ∇νV ≡ t ∂νV + t ΓνρV d = V µ +x ˙ νΓµ V ρ . (1.9) dλ νρ
Where ∇µ is the covariant derivative. Since t is a tangent to the curve, a vector field V on C for which ν µ µ t ∇νV = f(λ)V (1.10) with an arbitrary function f is said to be parallely transported along the curve. From (1.8), (1.9) and by taking f(λ) ≡ 0 it is then clear that a geodesic is curve whose tangent is parallely transported along it.
Now consider the infinitesimal transformation
xµ → x0µ = xµ − kµ(x) , (1.11) changing the action to Z b 0 q 0µ 0ν 0 S [x(λ)] = −mc dλ x˙ x˙ gµν . (1.12) a Under this transformation the metric transforms like
∂xλ ∂xσ g (x) → g0 (x0) = g (x) . (1.13) µν µν ∂x0µ ∂x0ν λσ From this it follows that
0 σ σ σ gµν(x) = gµν(x) + (k ∂σgµν + ∂νk gµσ + ∂µk gσν) (1.14)
= gµν(x) + Lkgµν(x) , (1.15) where
Lkgµν(x) = ∇µkν + ∇νkµ (1.16) is the Lie derivative of the metric.
The function x(λ) comes down to considering a line in spacetime within a certain coordi- nate patch. Relabeling the spacetime points by a coordinate transformation is independent of which specific line in spacetime is considered, so kµ(x) is independent of λ. With all this, the transformation of the action (1.12) becomes
Z b 0 µ ν −1/2 S [x(λ)] = S[x(λ)] − m dλ (−x˙ x˙ gµν) (Lkgµν) . (1.17) 2 a Thus the action is invariant if
Lkgµν = 0 . (1.18) Chapter 1. Classical aspects of black holes 4
A vector field kµ(x) satisfying this property is called a Killing vector field. It immediately follows from (1.16) that a Killing vector field satisfies
∇µkν = −∇νkµ , (1.19) which is the Killing vector lemma.
Interpreting vector fields as derivation operators on spacetime functions, a Killing vector field can be seen as the generator of a symmetry of the spacetime. To define what a spacetime symmetry is, first consider two (pseudo-) Riemannian manifolds (M, g) and (M 0, g0). A diffeo- 0 0 0 morphism f between M and M is an isometry if f∗g = g . If M = M than an isometry is called a symmetry of the spacetime. Hence, a Killing vector field is a vector field whose flow (= integral curves) consists of isometries.
Since for each Killing vector field there is a symmetry of the action, there also is a conserved charge. This charge is µ Q = k pµ (1.20) where pµ is the particle’s 4-momentum
∂L p = µ ∂x˙ µ dxν = m g when m 6= 0 . (1.21) dτ µν To see how this charge is conserved, use the fact that the action is invariant under the above coordinate transformations Z ∂L ∂L 0 = δS = dλ δxµ + δx˙ µ (1.22) ∂xµ ∂x˙ µ Z d ∂L ∂L = dλ δxµ + δx˙ µ (1.23) dλ ∂x˙ µ ∂x˙ µ Z d ∂L = dλ δxµ , (1.24) dλ ∂x˙ µ where the Euler-Lagrange equations (1.6) were used in the second step. From this follows
∂L ∂L δxµ = kµ = constant . (1.25) ∂x˙ µ ∂x˙ µ As mentioned before, a vector field is a derivation operator generally expressed as
µ k = k ∂µ . (1.26)
For any vector field k, local coordinates can be found such that
∂ k = , (1.27) ∂ξ Chapter 1. Classical aspects of black holes 5 where ξ is one of the coordinates. In such a coordinate system one has
∂ L g = g . (1.28) k µν ∂ξ µν
So k is a Killing vector field if gµν is independent of ξ. For example, in a static spacetime ∂tgµν = 0, so ∂/∂t is a Killing vector field. The corresponding conserved quantity is
dt mg = m e , (1.29) 00 dτ where e is the energy per unit mass.
1.2 Null surfaces
Let S(x) be a smooth function of the spacetime coordinates xµ and consider a family of hyper- surfaces S = constant. The vector fields normal to the hypersurfaces are
∂ l = −f(x)(gµν∂ S) , (1.30) ν ∂xµ where f(x) is an arbitrary non-zero spacetime function. If l is a null vector, i.e. l2 = 0, for a particular hypersurface N in the family, then N is said to be a null hypersurface.
A vector t is tangent to a hypersurface if t · l = 0. But for the null hypersurface N it holds that l · l = 0, so l is itself a tangent vector:
dxµ lµ = , (1.31) dλ for some null curve xµ(λ) in N .
These curves appear to have a very special property. This can be seen by considering the vector l · ∇lµ. By using the definition for the normal lµ and the fact that the covariant derivative of the metric is zero, one gets
µ µν l · ∇l = −l · ∇(f(x)g ∂νS) ρ µν µν ρ = − (l ∂ρf) g ∂νS − f g l ∇ρ∂νS, (1.32) which can be rewritten by using the definition of the normal lµ and the symmetry of the Levi- Civita connection µ µ µν ρ l · ∇l = (l · ∂ ln f) l − f g l ∇ν∂ρS. (1.33) Again making use of the definition of lµ in the second term, this becomes
µ µ ρ µ −1 l · ∇l = (l · ∂ ln f) l + l f∇ f lρ µ ρ µ µ 2 = (l · ∂ ln f) l + l ∇ lρ − (∂ ln f) l 1 = (l · ∂ ln f) lµ + ∂µl2 − (∂µ ln f) l2 . (1.34) 2 Chapter 1. Classical aspects of black holes 6
Where it was used in the last step that l2 is a scalar so that the covariant derivative can be replaced by a partial derivative.
When evaluating (1.34) on the null surface N ,(1.31) and l2 = 0 give
d 1 l · ∇lµ | = ln f lµ | + ∂µl2 | . (1.35) N dλ N 2 N
It does not follow that ∂µl2 is zero on N , unless the whole family of hypersurfaces S = constant 2 µ 2 is null. However, since l is constant on N , it holds that t ∂µl = 0 for any tangent vector t of N . Thus it follows that 2 ∂µl |N ∝ lµ , (1.36) and therefore one gets from (1.35) µ µ l · ∇l |N ∝ l . (1.37) From (1.10) it is clear that this expresses parallel transport of the tangent vector l. So xµ(λ) appears to be a geodesic. This can be made explicit by choosing the function f such that l · ∇lµ = 0 on N . Using (1.31) one gets
µ 0 = l · ∇l |N ν µ =x ˙ ∇νx˙ µ µ ν σ =x ¨ + Γνσx˙ x˙ , (1.38) which is the geodesic equation (1.8).
The null geodesics xµ(λ) for which the tangent vectorsx ˙ µ(λ) are normal to a null hypersurface N , are called the generators of N .
1.3 The Cauchy problem in General Relativity
In this thesis frequent use will be made of differential equations in a general spacetime. Here, a closer look is taken at the existence and uniqueness of solutions to such differential equations.
1.3.1 Differential equations in curved spacetime
Consider the curved spacetime version of the Klein-Gordon equation
µν 2 (g ∂µ∂ν − m )ψ = 0 , (1.39) where the Minkowski metric is replaced by a general spacetime metric according to the principle of minimal coupling, to be discussed in the next chapter. The Klein-Gordon equation will serve in this section as a representative of the entire class of second order hyperbolic equations.
One would like to obtain that, analogous to Minkowski spacetime, solutions of this Klein- Gordon are uniquely determined by some initial conditions on a spacelike hypersurface. In Chapter 1. Classical aspects of black holes 7 other words, embedding this 3-dimensional hypersurface containing the initial data in the 4- dimensional spacetime (M, gµν) should, together with the Klein-Gordon equation, result in a unique spacetime function ψ on M.
In an arbitrary curved spacetime, the classical existence and uniqueness properties of solu- tions to equation (1.39) can be very different from that of Minkowski spacetime. Two examples will illustrate this point: 1) Let the spacetime be a flat 4-torus, with spatial periodicity L and time periodicity T such that T 2/L2 is irrational (T and L do not have to be integers). Then in this spacetime the massless Klein-Gordon has as only solution ψ = constant [2]. 2) Consider any spacetime with a timelike singularity, such as Minkowski spacetime with a timelike line removed, or the Schwarzschild solution with negative mass. Since equation (1.39) does not restrict what can emerge from a singularity, there is no possibility that uniqueness can hold for the solutions to (1.39) with given initial conditions on a spacelike hypersurface.
From this examples it is clear that there exists the necessity for a criterion to determine whether or not a spacetime permits the existence and uniqueness of solutions of second order hyperbolic equations, given some initial data on a hypersurface.
1.3.2 Global Hyperbolicity
There is a simple condition on a spacetime (M, gµν) which guarantees that differential equa- tions have a well posed initial value formulation. First, it is necessary to restrict the attention to spacetimes which are time orientable, which means that a continuous choice can be made throughout the spacetime of which half of each light cone constitutes the ’future’ direction and which half consitutes the ’past’.
Now, let Σ ⊂ M be any closed set of points which is achronal, i.e. no pair of points p, q ∈ Σ can be joined by a timelike curve. The domain of dependence of Σ is defined by
D(Σ) = {p ∈ M| every (past and future) inextendible causal curve through p intersects Σ} (1.40)
If D(Σ) = M, then Σ is said to be a Cauchy surface for the spacetime (M, gµν). A Cauchy surface is then automatically a 3-dimensional hypersurface. If a spacetime admits a Cauchy surface it is said to be globally hyperbolic.
There is a very important theorem regarding the structure of globally hyperbolic spacetimes [3]:
If (M, gµν) is globally hyperbolic with Cauchy surface Σ, then M has topology R ⊗ Σ. Fur- thermore, M can be foliated by a one-parameter family of smooth Cauchy surfaces Σt, which means that a smooth ’time coordinate’ t can be chosen on M such that each surface of constant t is a Cauchy surface.
Since every causal curve intersects a Cauchy surface in a unique point and causal curves do not intersect with other causal curves, one might expect to have a well defined deterministic Chapter 1. Classical aspects of black holes 8 classical evolution from initial conditions given on Σ. That this is indeed the case for (1.39) is stated in the following theorem [1]:
Let (M, gµν) be a globally hyperbolic spacetime with Cauchy surface Σ. Then the Klein-Gordon equation has a well posed initial value formulation in the following sense: Given any pair of ∞ smooth C -functions (ψ0, ψ˙0) on Σ, there exists a unique solution ψ to (1.39), defined on all µ µ of M, such that on Σ one has ψ = ψ0 and n ∇µψ = ψ˙0 where n denotes the unit, future directed normal to Σ. Furthermore, for any closed subset S ⊂ Σ, the solution ψ restricted to D(S) depends only upon the initial data on S. In addition, ψ is smooth and varies continuously with the initial data.
Similar results hold for a much more general class of linear wave equations and systems of linear wave equations. The above theorem also continues to hold if a smooth source term f is inserted to the right hand side of (1.39). It then follows directly from the domain of depen- dence property of this theorem that there exists a unique advanced and retarded solution to the Klein-Gordon equation with source term. This means that in a globally hyperbolic space- time there exist a unique advanced and retarded Green’s function for the Klein-Gordon equation.
For the remainder of this thesis, it is important to realize that a black hole spacetime is globally hyperbolic [2].
1.4 The Horizon
A basic phenomenon which underlies most of the discussions in further chapters is that of a massive object or cloud collapsing to a black hole. The necessary background on this process is given and the concept of a black hole horizon is introduced in the first part of this section. When doing quantum field theory in the gravitational collapse spacetime an intriguing phe- nomenon, the Hawking radiation, occurs, resulting from the special causal structure of black hole spacetimes. For that reason, the causal structure of black hole spacetimes is discussed in the second and third part of this section.
1.4.1 Gravitational collapse
Outside a static spherically symmetric object such as a star, the solution of the Einstein equa- tions is given by the Schwarschild metric [4]:
2GM 2GM −1 ds2 = 1 − dt2 − 1 − dr2 − r2dΩ2 . (1.41) r r
This solution holds for r greater than some r0 which corresponds to the surface of the star and is joined for r < r0 onto a solution which depends in detail on the radial distribution of density and pressure in the object. Now Birkhoff’s theorem states that the outside solution will still be part of the Schwarzschild solution cut off by the surface of the object even if this object is not static, provided that it remains spherically symmetric. This can be rigorously proven [1]. Chapter 1. Classical aspects of black holes 9
If the star is static then r0 must be greater than the Schwarschild radius 2GM. This fol- lows because the surface of a static star at r0 must correspond to the orbit of a timelike Killing vector field, and in the Schwarschild solution there is a timelike Killing vector field k only where r > 2GM because ∂ 2GM k = ⇒ k2 = g = 1 − , (1.42) ∂t tt r 2 so k is positive only for r > 2GM. If r0 were less than 2MG, the surface would be expanding or contracting.
The process of black hole formation is generally known. A star’s nuclear fuel gets exhausted, it will cool and the pressure will be reduced, and so it will contract. Now suppose that this con- traction cannot be halted by the pressure before the radius becomes less than the Schwarschild radius, which seems likely for stars of greater than a certain mass [1]. Then since the solution outside the star is the Schwarschild solution, there will be a closed trapped surface S around the star. By a closed trapped surface is meant a closed spacelike two-surface such that the two families of null geodesics orthogonal to S are converging at S. For a more formal definition, consider a two-dimensional, closed, convex, spacelike surface S in a curved spacetime. Let A be the surface area of S (calculated using the induced metric on S). Define a time coordinate t such that t = 0 on that surface. Suppose that the surface at t = 0 divides 3-space into two regions: an outer region V1 and an inner region V2. A small instant later, at time t = , 3-space is divided into three regions:
1) an outer region V1 that is spacelike separated from S, 2) an inner region V2 that is also spacelike separated from S, 3) a region V3 between V1 and V2 that can be reached by timelike geodesics from S. Its boundary can be reached with light-like geodesics from S.
Let S1 be the boundary between V1 and V3 and S2 be the boundary between V2 and V3. The surfaces S1 and S2 have areas A1 and A2. This situation is sketched on figure 1.2.
Figure 1.2: 3-space divided in 3 parts by signals moving inwards and outwards of a surface S over a time . Chapter 1. Classical aspects of black holes 10
Now, define the expansion rates θ1, θ2 of these two surfaces as follows dA dA θ = 1 , θ = 2 (1.43) 1 d 2 d Under non-exotic circumstances, such as in a flat spacetime, certainly the outer surface expan- sion rate is positive: θ1 > 0. The inner one is usually negative. However, inside a black hole, one can have a trapped surface. S is called trapped if θ1 and θ2 are negative or zero. A surface is marginally trapped if both expansion rates are strictly equal to zero. For a pure Schwarzschild black hole, the surface r = 2M is marginally trapped [5].
That there exists something like a closed trapped surface can be seen by (1.42). This equation implies that the Killing vector field generating time translations for distant observers becomes spacelike at r = 2MG, so moving forward in time for distant observers becomes moving forward in space on the inside of the closed trapped surface. One may think of S as being in such a strong gravitational field that even the ’outgoing’ light rays are dragged back and are, in fact, converging. In that case the singularity theorems of Hawking and Penrose [1] imply that a sin- gularity will occur provided that causality is not violated and the appropriate energy condition holds. Of course, here, because the exterior solution is the Schwarzschild solution, it is obvious that there must be a singularity.
The two principal reasons why a star may depart from spherical symmetry are that it may be rotating of may have a magnetic field. One may get some idea of how large the rotation may be without preventing the occurence of a trapped surface by considering the Kerr solution (see below). This solution can be thought of as representing the exterior solution for a body with mass M and angular momentum J = aGM. If a is less than GM there are closed trapped surfaces, buf if a is greater than GM they do not occur. Thus one might expect that if the angular momentum of the star were greater than the square of its mass, it would be able to halt the contraction of the star before a closed trapped surface developed. Another way of seeing this is that if J = (GM)2 and angular momentum is conserved during the collapse, then the velocity of the surface of the star would be about the velocity of light when the star was at its Schwarzschild radius. Now many stars have an anglar momentum greater than the square of their mass. However, it seems reasonable to expect some loss of angular momentum during the collapse because of braking by magnetic fields and because of gravitational radiation [1]. The situation is therefore that in some stars, and probably most, angular momentum would not prevent occurance of closed trapped surfaces, and hence a singularity. It also appears that the rate of increase of the magnetic pressure is too slow to have a significant effect on the collapse.
Now consider a body of 108 solar masses. If this collapsed to its Schwarzschild radius, the density would only be of the order of 10−4g/cm3, which is less than the density of air [1]. If the matter were fairly cold initially, the temperature would not have risen sufficiently either to support the body or to ignite the nuclear fuel. This example shows that the conditions when a body passes through its Schwarzschild radius need not be in any way extreme.
There are several models of gravitational collapse where the dynamics can be calculated an- alytically. Examples are: the collapse of a ball of pressureless dust with uniform density and the case of spherical symmetric collapse with internal pressure [6–8]. In the first example, the Chapter 1. Classical aspects of black holes 11 solution of the Einstein equations inside the dust ball consists of a portion of a closed Friedman- Lemaˆıtreuniverse which is glued to the Schwarzschild solution on the outside of the dust ball. This feature should be no surprise since the Friedman-Lemaˆıtresolution is defined to describe the inside of a homegeneous and isotropic mass density.
But the most instructive model of gravitational collapse is that of a thin contracting spher- ical shell of massless particles with total energy M. Inside the shell, before the passage of the particles, the spacetime is flat and thus Minkowski. Again by Birkhoff’s theorem, the spacetime outside the shell will be the Schwarzschild spacetime. The closed trapped surface will form at the origin of 3-space with its radius increasing in time up to the point it intersects with the contracting shell and from that point on, it remains a constant surface in 3-space [9]. This is depicted in figure 1.3. There is evidently no local quantity in the Minkowski spacetime inside the shell that will distinguish the presence of the closed trapped surface whose occurence is due entirely to the future collapse of the shell.
Figure 1.3: The formation of a closed trapped surface in a collapsing shell of massless particles.
As another example, imagine a person hovering safely a certain distance above the closed trapped surface of a black hole of mass M. But then, an object of considerable mass M 0 falls into the black hole, thereby increasing its mass to M +M 0. As a result of this, the closed trapped surface will increase its radius, thereby trapping the at first perfectly safe person.
Now the horizon of a black hole is defined as the boundary of the region of space- time from which no signal can escape to infinity. It is a surface that seperates spacetime in an outer and an inner region. Any light ray which orginates in the inner region can never reach any point on the outer region. Based on the two examples above, it is clear that the horizon is a truely global phenomenon since its location is determined by all future events. This will become more explicit when considering the Penrose diagram of gravitational collapse to a black hole later on. Chapter 1. Classical aspects of black holes 12
Figure 1.4: A person getting trapped behind the horizon.
By considering the Schwarzschild metric (1.41) it looks like the horizon is singular because of the divergence of grr as r → 2MG. However, this is not the case, as it will appear that this divergence is only a singularity of the Schwarzschild coordinates, just like the spherical coordinates fail to describe the north and south pole of a 2-sphere. It is not a singularity of the spacetime manifold. In the next two paragraphs we will investigate what exactly is going on in the region around the horizon, with a special emphasis on the causal structure of the spacetime, by considering an outside and an infalling observer.
1.4.2 Outside observers and the end of time
Consider two observers in Schwarzschild spacetime who are stuck at fixed spatial coordinate values (r1, θ1, φ1) and (r2, θ2, φ2). Then, the proper time of observer i will be related to the coordinate time t by dτ 2GM 1/2 i = 1 − . (1.44) dt ri
Suppose that the observer O1 emits a light pulse which travels to observer O2, such that O1 measures the time between two succesive crests of the light wave to be ∆τ1. Each crest follows the same path to O2, except that they are separated by a coordinate time
2GM −1/2 ∆t = 1 − ∆τ1 . (1.45) r1
This separation in coordinate time does not change along the photon trajectories, but the second observer measures a time between successive crests given by
2GM 1/2 ∆τ2 = 1 − ∆t r2 1/2 1 − 2GM/r2 = ∆τ1 . (1.46) 1 − 2MG/r1 Chapter 1. Classical aspects of black holes 13
Since these intervals ∆τi measure the proper time between two crests of an electromagnetic wave, the observed frequencies will be related by
ω ∆τ 2 = 1 ω1 ∆τ2 1 − 2GM/r 1/2 = 2 . (1.47) 1 − 2MG/r1
This is an exact result for the frequency shift.
Now consider radial null curves, thus those for which θ and φ are constant and ds2 = 0.
2GM 2GM −1 0 = 1 − dt2 − 1 − dr2 , (1.48) r r from which follows that dt 2GM −1 = ± 1 − . (1.49) dr r This measures the slope of the light cones on a spacetime diagram of the t − r plane. For large r the slope is ±1 as in Minkowski spacetime, while as one approaches r = 2MG one gets dt/dr → ±∞, and the light cones ’close up’. This is depicted on figure 1.5.
Figure 1.5: The closing up of the lightcones in Schwarschild spacetime.
Thus a light ray which approaches r = 2MG never seems to get there, at least in this coordinate system, instead it seems to asymptote to this radius. This is the description for a distant ob- server, collecting information available through his past light cone. It is now clear that such an observer never actually sees events on the horizon. In this sense the horizon must be regarded as at the end of time. It also follows from (1.47) that any particle or wave which falls through the horizon is seen by the distant observer as asymptotically approaching the horizon as it is infinitely redshifted.
Before the mid-1960s the object that is now called a ’black hole’ was referred to in the En- glish literature as a ’collapsed star’ and in the Russian literature as a ’frozen star’ [10]. The corresponding mental picture, based on stellar collapse as viewed in Schwarzschild coordinates as described above, was one of a collapsing star that contracts more and more rapidly as the grip of gravity gets stronger and stronger, the contraction then slowing because of a growing gravitational redshift and ultimately freezing to a halt at an ’infinite-redshift’ surface at the Schwarzschild radius, there to hover for all eternity. From the work of Oppenheimer and Snyder Chapter 1. Classical aspects of black holes 14
[7] there was the awareness of an alternative viewpoint, that of an observer on the surface of the collapsing star who sees no freezing but instead experiences collapse to a singularity in a painfully short time. But because nothing inside the infinite-redshift surface can ever influence the external universe, that ’comoving viewpoint’ seemed irrelevant for astrophysics. Thus as- trophysical theorizing in the early 1960s was dominated by the ’frozen-star viewpoint’. As long as this viewpoint prevailed, physicists failed to realize that black holes can be dynamical, evolv- ing, energy-storing and energy-releasing objects (see section 1.8). Objects capable of colliding, vibrating widley and emitting huge bursts of gravitational waves [10].
1.4.3 Infalling observers and the equivalence principle
The fact that an outside observer never sees objects reach r = 2GM is a meaningful statement, but the fact that their trajectories in the t − r plane never reaches there is not. It is highly dependent on the coordinate system, and it is better to ask a more coordinate-independent question such as; do the infalling objects reach the Schwarzschild radius in a finite amount of proper time? The best way to do this is to change coordinates to a system which is better behaved at r = 2GM.
As a first step, the tortoise coordinates are introduced. The change to tortoise coordinates from the convential spherical coordinates is made by a change of the radial coordinate that maps the horizon to minus infinity, so that the resulting coordinate system covers only the region r > 2MG. The tortoise coordinate r∗ is defined by
1 2MG dr2 = 1 − (dr∗)2 , (1.50) 2MG r 1 − r and is explicitely given by r − 2MG r∗ = r + 2MG ln . (1.51) 2MG In tortoise coordinates the Schwarzschild metric takes the form 2MG ds2 = 1 − [dt2 − (dr∗)2] − r2[dθ2 + sin2 θdφ2] . (1.52) r
Where r is to be thought of as a function of r∗. The interesting point is that the radial-time part of the metric is now conformally flat. A space is called conformally flat if its metric can be brought to the form 2 µ ν ds = F (x)dx dx ηµν , (1.53) with ηµν being the conventional Minkowski metric. Actually, any two-dimensional space is conformally flat, so a slice through Schwarschild spacetime at fixed θ, φ is no exception. The transition to tortoise coordinates represents some progress, since the light cones now don’t seem to close up, furthermore, none of the metric coefficients becomes infinite at r = 2MG.
The next step is to define coordinates which are more naturally adapted to null geodesics. Chapter 1. Classical aspects of black holes 15
Put
u = t − r∗ (1.54) v = t + r∗ . (1.55)
Then outgoing radial null geodesics are characterized by u = constant and infalling ones satisfy v = constant. Now consider going back to the original radial coordinate r, but replacing the timelike coordinate t with the new coordinate v. These coordinates are known as Eddington- Finkelstein coordinates, in terms of which the metric becomes
2MG ds2 = 1 − dv2 − 2dvdr − r2dΩ . (1.56) r
Here, there is a first sign of progress. The determinant of the metric is
g = −r4 sin2 θ , (1.57) which is perfectly regular at r = 2GM. Therefore, the metric is invertible and it is clear once and for all that r = 2GM is simply a coordinate singularity in the original system (t, r, θ, φ). In the Eddington-Finkelstein coordinates the condition for radial null curves is solved by ( dv 0 (infalling) = (1.58) dr 2GM −1 2 1 − r (outgoing)
One can therefore see what happens: in this coordinate system the light cones remain well- behaved at r = 2MG, and this surface is at finite coordinate value. There is no problem in tracing the paths of null or timelike particles past the horizon. On the other hand, something interesting is certainly going on. Although the light cones don’t close up, they do tilt over, such that for r < 2GM all future-directed paths are in the direction of decreasing r. It is this aspect of the causal structure of black hole spacetimes that is so special. One could see this as if space and time ’have switched roles’. It is this effect that already was anticipated on by (1.42). This tilting of the light cone is illustrated in figure 1.6.
Figure 1.6: Tilting of the light cone in Eddington-Finkelstein coordinates.
The complete picture of the causal structure around a Schwarschild horizon, with the closing up and the tilting of the light cone is given in figure 1.7. Chapter 1. Classical aspects of black holes 16
Figure 1.7: The causal structure around the Schwarschild horizon.
It is comforting to have found this second point of view. Because one of the cornerstones of general relativity, the equivalence principle, states that a freely falling observer locally observes Minkowski spacetime, so he should not feel anything strange when passing the horizon. It is only at the singularity that things start to go terribly wrong.
1.5 The maximal extension of Schwarzschild spacetime
In section 1.2.3 the original coordinate t was changed to the new one v, which had the nice property that if one decreases r along a radial null curve v = constant, one goes right through the event horizon without any problems. From this it is clear that the initial coordinate system didn’t do a good job of covering the entire manifold. The region r < 2GM should certainly be included in the spacetime, since physical particles can easily reach there and pass through. However, there is no guarantee that now the entire spacetime is covered, perhaps there are other directions in which the manifold can be extended.
Notice that in the (v, r) coordinate system the event horizon can be crossed on future-directed paths, but not on past-directed ones. This seems unreasonable, since the Schwarzschild solution is time-independent. But u could have been chosen as coordinate instead of v, in which case the metric would have been 2GM ds2 = 1 − du2 + (dudr + drdu) − r2dΩ2 . (1.59) r Chapter 1. Classical aspects of black holes 17
Now one can once again pass through the event horizon, but this time only along past-directed curves. This means that one can consistently follow either future-directed or past-directed curves through r = 2MG but arrive at different places. This was to be expected since from the definitions (1.54) and (1.55) it follows that if v is constant and r decreases that t → +∞, while if u is constant and r decreases, then t → −∞ because the tortoise coordinate goes to −∞ as r → 2GM. Therefore, the spacetime is extended in two different regions, one to the future and one to the past.
Figure 1.8: Crossing the horizon at constant u along past directed curves.
The next step would be to follow space-like geodesics and see if more regions are uncovered. But a shortcut to the process is by defining coordinates that are good all over. A first guess might be to use both u and v at once instead of t and r, which leads to
1 2GM ds2 = 1 − (dvdu + dudv) − r2dΩ2 , (1.60) 2 r with r defined implicitely in terms of v and u by
1 (v − u) = r∗ . (1.61) 2 With this, the degeneracy with which we started out is re-introduced, in these coordinates r = 2MG is infinitely far away, at either v = −∞ or u = +∞. The thing to do is to change coordinates which pull these points into finite coordinate values, a good choice is u U = − exp(− ) (1.62) 4GM v V = exp( ) , (1.63) 4GM which in terms of the original (t, r) coordinates is
r 1/2 U = − − 1 e(r−t)/4GM (1.64) 2GM r 1/2 V = − 1 e(r+t)/4GM . (1.65) 2GM In these coordinates the Schwarzschild metric is
32G3M 3 ds2 = e−r/2GM dUdV − r2dΩ2 . (1.66) r Chapter 1. Classical aspects of black holes 18
Both U and V are null coordinates or ’radial light-like’ variables, in the sense that their corre- sponding vector fields ∂/∂U and ∂/∂V are null vectors. They are called the Kruskal-Szekeres coordinates in their light cone variant.
The surfaces of constant r > 2GM are timelike hyperbolas in sector I of figure 1.9, given by UV = negative constant . (1.67)
As r tends to 2MG the hyperbolas become the boken straight lines H+ and H− which are called the extended past and future horizons and satisfy
UV = 0 . (1.68)
Although the extended horizons lie at finite value of the Kruskal-Szekeres coordinates, they are located at Schwarzschild time ±∞. So a particle trajectory which crosses H+ in a finite proper time, crosses r = 2GM only after an infinite Schwarzschild time.
The region r < 2MG is region II on figure 1.9. In this region the surfaces of constant r are the space-like hyperboloids given by
UV = positive constant . (1.69)
The singularity at r = 0 occurs at UV = 1 . (1.70)
Figure 1.9: The maximal extended Schwarzschild spacetime in Kruskal coordinates. Chapter 1. Classical aspects of black holes 19
Now it is important to realize that equations (1.67), (1.68), (1.69) and (1.70) refer to a bigger coordinate interval than used above. The Kruskal-Szekeres coordinates should be allowed to range over every value they can take without hitting the singularity at r = 0. It is clear from figure 1.9 that up to now, only positive V are considered. By adding also the negative values for V , regions III and IV are introduced on figure 1.9 and the maximal extension for the Schwarzschild geometry is obtained.
The Kruskal-Szekeres coordinates have the nice property that outgoing radial null geodesics are given by U = constant and incoming radial null geodesics by V = constant. So all light rays and timelike trajectories lie within a two-dimensional light cone bounded by 45◦ lines in figure 1.9. Also a nonradially directed light ray or time-like trajectory always lies inside the two-dimensional light cone. With this in mind, it is easy to understand the causal properties of the maximal analytic extension of the Schwarzschild geometry in figure 1.9. An observer in region II can send signals across H+ to region I and also to infinity. All signals sent in region II must eventually hit the future singularity. From region III no signal can ever get to region I, so it is also behind the horizon. On the other hand, observers in region IV can communicate with region I by sending signals across H−. Region I however, cannot communicate with region IV . All of this is usually described by saying that regions II and III are behind the future horizon while regions III and IV are behind the past horizon. Region IV is the time-reverse of region II, the black hole, and is a part of spacetime from which things can escape to us, while nothing can get in. This time-reversed black hole is called a white hole. There is a singularity in the past, out of which the universe appears to spring. Region III is another asymptotically flat region of spacetime, a mirror image of ours that is not able to communicate with us at region I either forward or backward in time.
1.6 Killing horizons and surface gravity
The concept of a Killing horizon is of vital importance for the discussion of black holes:
Killing horizon A null hypersurface N is a Killing horizon of a Killing vector field ξ if on N , ξ is normal to N .
Now let the vector field l be the normal to N . In section 1.2 it was shown that l can be chosen such that l · ∇lµ = 0. Since N is a Killing horizon of the Killing vector field ξ, one then has ξ = f˜ l (1.71) for some spacetime function f˜. So it follows that
µ σ µ ξ · ∇ξ = f˜ l ∇σ(f˜ l ) σ µ σ µ = f˜ l (∂σf˜)l + f˜ l f˜∇σl = f˜ l · ∂f˜ lµ = ξ · ∂ ln|f˜| ξµ , (1.72) Chapter 1. Classical aspects of black holes 20 where κ = ξ · ∂ ln|f˜| is called the surface gravity.
This definition determines κ up to a constant factor. Since ξ2 = 0 on the null surface N , there is no natural normalization for ξ. But in an asympotically flat spacetime there is a nat- ural normalization at spatial infinity. For example, for a time-translation Killing vector field k one can choose k2 → 1 as r → +∞ . (1.73)
This fixes k, and hence κ, up to a sign. The sign of κ is fixed by requiring k to be future-directed.
Since ξ is hypersurface orthogonal at the horizon, by Frobenius’s theorem (see appendix A), one has
ξ[µ∇νξσ] = 0 . (on the horizon) (1.74)
Using the Killing vector lemma ∇νξσ = −∇σξν, this implies
ξσ∇µξν = −2ξ[µ∇ν]ξσ (1.75) on the horizon. Contracting with ∇µξν, one finds
µ ν µ ν ξσ(∇ ξ )(∇µξν) = −2(ξµ∇ ξ )(∇νξσ) ν = −2κξ ∇νξσ 2 = −2κ ξσ . (1.76)
Thus, one obtains a simple explicit formula for κ,
1 κ2 = − (∇µξν)(∇ ξ ) , (1.77) 2 µ ν where evaluation on the horizon is understood. This equation provides us with a physical interpretation of κ in the following way. One has everywhere, i.e. not just on the horizon,
[µ ν σ] µ ν σ µ ν σ 3(ξ ∇ ξ )(ξ[µ∇νξσ]) = ξ ξµ(∇ ξ )(∇νξσ) − 2(ξ ∇ ξ )(ξν∇µξσ) . (1.78)
Since ξ[µ∇νξσ] = 0 on the horizon, the gradient of the left-hand side vanishes on the horizon. µ On the other hand, by (4.162), ∇ν(ξ ξµ) does not vanish when κ is not equal to zero. Hence, µ by l’Hospital’s rule, the left side of equation (4.148) divided by ξ ξµ must approach zero on the horizon. Thus, using (4.151), one finds
2 ν σ µ ν κ = lim[−(ξ ∇νξ )(ξ ∇µξσ)/ξ ξν] , (1.79) where ’lim’ stands for the limit approaching the horizon. Now, a particle on a time-like orbit of a Killing vector field ξ has a 4-velocity
ξµ uµ = , (1.80) |ξ| Chapter 1. Classical aspects of black holes 21 since u ∝ ξ and u · u = 1. Therefore, its proper four-vector acceleration is
aµ = u · ∇uµ ξ · ∇ξµ (ξ · ∂ξ2)ξµ = − . (1.81) ξ2 2ξ2
2 µ ν But for a Killing vector field, ξ · ∂ξ = 2ξ ξ ∇µξν = 0 by the Killing vector lemma, so
ν σ σ ξ ∇νξ a = µ (1.82) ξ ξµ in (1.79) is just the proper acceleration on an orbit of ξ. Thus, we find
κ = lim(|ξ|a) , (1.83) √ σ p µ where a = −a aσ and |ξ| = ξ ξµ. In the case of a static black hole, one has ξ = k. Then |ξ| is just the redshift factor, and |ξ|a is the force that must be exerted at infinity to hold a unit test mass in place [11]. Thus, κ is the limiting value of this force at the horizon, which explains the name surface gravity. (Of course, the locally exerted force a becomes infinite at the horizon.) As we will see later, for a rotating black hole, a test mass cannot be held stationary with respect to infinity near the black hole, but one continues to refer to κ as the surface gravity.
It can be shown for a general Killing horizon that κ is constant on orbits of ξ [12]. Now suppose that κ 6= 0 on one orbit of ξ in N . Then this orbit coincides with only a part of a null generator of N . To see this, one can choose coordinates on N such that
∂ ξ = (1.84) ∂α at all points where ξ 6= 0. This means that the group parameter α is one of the coordinates. Then if α = α(λ) on an orbit of ξ with an affine parameter λ, one has
dλ d ξ| = = f˜ l . (1.85) orbit dα dλ So dλ d dxµ(λ) f = and l = = ∂ . (1.86) dα dλ dλ µ Then, the surface gravity κ is by definition
∂ κ = ln|f˜| , (1.87) ∂α
κα where κ is constant on an orbit on N . Thus, for such orbits, f = f0e for an arbitrary constant f0. Because of the freedom to shift α by a constant, one can choose f0 = ±κ without loss of generality. This choise implies
dλ = ±κeκα ⇒ λ = ±eκα , (1.88) dα where the integration constant was chosen to be zero. As α ranges from −∞ to +∞, one covers either the portion λ > 0 of the generator of N , or the portion λ < 0, depending on the sign Chapter 1. Classical aspects of black holes 22 choise in (1.88). The bifucation point λ = 0 is a fixed point of ξ, which can be shown to be a 2-sphere, called the bifurcation 2-sphere. A Killing horizon satisfying these properties is called a bifurcate Killing horizon. The general structure of a bifurcate Killing horizon is given on figure 1.10.
Figure 1.10: A bifurcate Killing horizon.
Now consider the Schwarzschild metric in ingoing Eddington-Finkelstein coordinates (1.56) as derived in section 1.4.3. The vector field normal to the family of surfaces S = r = constant according to (1.30) is given by
2MG ∂S ∂ ∂S ∂ ∂S ∂ l = f(r) 1 − + + r ∂r ∂r ∂r ∂v ∂v ∂r 2MG ∂ ∂ = f(r) 1 − + . (1.89) r ∂r ∂v
From which follows
2 µν 2 l = g ∂µS ∂νSf = grrf 2 2GM = 1 − f 2 . (1.90) r
So the horizon r = 2MG is a null surface, with normal
∂ l| = f . (1.91) r=2MG ∂v In Kruskal-Szekeres coordinates the horizon is given by U = 0. The vector field normal to the family of surfaces U = constant is
f r ∂ ˆl = er/2M , (1.92) 32M 3 ∂V Chapter 1. Classical aspects of black holes 23 where (1.66) was used. So at the horizon this becomes
f e ∂ ˆl | = . (1.93) N 16M 2 ∂V
Because gVV is zero in the Krukal-Szekeres metric, ˆl2 is identically zero, so U = constant is a null surface for any constant. It also follows that ∂µˆl2 is zero, so (1.35) implies that ˆl · ∇ˆlµ = 0 if f is constant. By choosing f = 16M 2e−1 the normal to the horizon U = 0 becomes
∂ ˆl = . (1.94) ∂V Now take ξ = k, with k the time-translation Killing vector field of the stationary black hole spacetime as normalized above. Making use of (1.65), k can be expressed in terms of the Kruskal-Szekeres coordinates as ∂ ∂U ∂ ∂V ∂ k = = + ∂t ∂t ∂U ∂t ∂V 1 ∂ 1 ∂ = − U + V , (1.95) 4MG ∂U 4MG ∂V Where only region I of the Kruskal-Szekeres spacetime is considered (see figure 1.9). So on the future horizon U = 0, k is given by
1 ∂ k = V = f˜ˆl , (1.96) 4MG ∂V where it follows from (1.94) that 1 f˜ = V. (1.97) 4MG So the horizon U = 0 is a Killing horizon of k. The surface gravity is
1 ∂ 1 κ = k · ∂ ln|f˜| = V ln| V | 4MG ∂V 4MG 1 = . (1.98) 4MG
Reinstalling factors of c, the surface gravity of a Schwarzschild black hole is κ = c3/4GM.
1.7 Penrose diagrams
When considering problems involving black holes frequent use is made of the so-called Penrose diagrams, which are a very convenient schematic way to represent spacetimes. In this section the main features of Penrose diagrams are presented, with an emphasis on the specific examples to be used later in this thesis.
1.7.1 Conformal compactification
A Penrose diagram is basically obtained by performing two subsequent transformations (each one possibly in multiple steps). The first one, which is not always necessary, is a coordinate Chapter 1. Classical aspects of black holes 24 transformation ensuring that radial null geodesics lie at ±45◦. The second is a conformal trans- formation which respects angles but changes distances. The purpose of this second transfor- mation is to bring infinity at finite distance so that a compact representation of the spacetime can be made. To be more precise, all the points at infinity in the original metric should be at a finite affine parameter value in the new metric. The recipe can’t be made more specific because it varies for different spacetimes. This will be illustrated by the following two examples.
Minkowski spacetime Take the Minkowski metric in spherical coordinates
ds2 = dt2 − dr2 − r2dΩ2 . (1.99)
Radial light rays propagate on the light cone dt ± dr = 0. This means that the first coor- dinate transformation putting radial null geodesics at ±45◦ is not necessary here. Under the transformation
u0 = t − r v0 = t + r (1.100) the Minkowksi metric becomes 1 ds2 = du0dv0 + (u0 − v0)2dΩ2 . (1.101) 4 Now set π π u0 = tan ω − < ω < 2 2 π π v0 = tan η − < η < , (1.102) 2 2 where η ≥ ω since r ≥ 0. In these coordinates the metric is
ds2 = (2 cos ω cos η)−2[4dω dη − sin2(η − ω)dΩ2] . (1.103)
To approach infinity in this metric one must take |ω| → π/2 or |η| → π/2, so by taking
Λ = 2 cos ω cos η (1.104) these points are brought to finite affine parameter in the new metric obtained by the conformal transformation ds˜2 = Λ2ds2 = 4dω dη − sin2(η − ω)dΩ2 . (1.105)
Now the points at infinity can be added. Taking the restriction η ≥ ω into account, these are ( ( ( ω = −π/2 u0 → −∞ r → +∞ 1) ⇔ ⇔ ⇒ spatial infinity i0 (1.106) η = π/2 v0 → +∞ t is finite Chapter 1. Classical aspects of black holes 25
( ( ( ω = ±π/2 u0 → ±∞ r is finite 2) ⇔ ⇔ ⇒ future/past timelike infinity i± η = ±π/2 v0 → ±∞ t → ±∞ (1.107)
( ( r → +∞ ω = −π/2 u0 → −∞ 3) ⇔ ⇔ t → −∞ ⇒ past null infinity I− (1.108) η 6= π/2 v0 is finite r + t is finite
( ( r → +∞ ω 6= +π/2 u0 is finite 4) ⇔ ⇔ t → +∞ ⇒ future null infinity I+ (1.109) η = +π/2 v0 → +∞ r − t is finite
These points together form conformal infinity. They are not part of the original spacetime. Minkowski spacetime is now conformally embedded in the new spacetime described by the met- ric ds˜2 with boundary at Λ = 0.
Introducing the new time and space coordinates τ, χ by
τ = η + ω χ = η − ω (1.110) the metric becomes ds˜2 = Λ2ds2 = dτ 2 − dχ2 − sin2 χdΩ2 , (1.111) with Λ = cos τ + cos χ.
The orginal coordinates (t, r) are related to (τ, χ) by
1 1 2t = tan( (τ + χ)) + tan( (τ − χ)) 2 2 1 1 2r = tan( (τ + χ)) − tan( (τ − χ)) (1.112) 2 2 χ is an angular variable which must be identified modulo 2π. If no other restriction is placed on the ranges of τ and χ, this metric ds˜2 is that of the Einstein static universe, which has the 3 topology R(time)⊗S (space). The 2-spheres of constant χ 6= 0, have radius |sin χ| (the points χ = 0, π are the poles of the spherical coordinate system describing a 3-sphere). The entire 3 manifold R ⊗ S can be drawn as a cylinder, in which each circle is a 3-sphere. Each point (τ, χ) on the cylinder represents the half of a 2-sphere, where the other half is the point (τ, −χ). This is represented in figure 1.11. The shaded region represents the sector corresponding to −π ≤ τ + χ ≤ π and −π ≤ τ − χ ≤ π as follows from (1.114) and (1.110). Chapter 1. Classical aspects of black holes 26
Figure 1.11: Conformal compactified Minkowski spacetime embedded in the Einstein static universe.
The restriction r ≥ 0 now becomes χ ≥ 0. This should not lead to the wrong conclusion, it is indeed the entire shaded region that represents the compactified Minkowski spacetime, and not 3 only half of it. The way the topology R ⊗ S is depicted is invariant under χ → −χ. Restricting χ to be positive justs picks one of these two possible representations. It are indeed the entire 2-spheres, represented by opposite points on the circles that make up Minkowski spacetime. So actually, the restriction is fulfilled quite naturally.
Identifying opposite points as one two-sphere of surface 4π sin2 χ and representing them by a single point results in a triangle representing Minkowski spacetime, as seen in figure 1.12. This is the Penrose diagram of Minkowski spacetime. As a result of this construction every point represents a 2-sphere, except i0, i+, i− and the r = 0 line.
Spatial sections of the compactified spacetime are topologically S3 because of the addition of the point i0. Thus, they are compact but have no boundary. This is not true for the whole spacetime. Asymptotically it is possible to identify points on the boundary of the compactified spacetime to obtain a compact manifold without boundary. In general, this is not possible because i+ and i− can be singular points which cannot be added. This will be the case in the next example.
Schwarzschild spacetime Take the metric of Schwarzschild spacetime in the form (1.60) as derived in section 1.3:
1 2GM ds2 = 1 − (dvdu + dudv) − r2dΩ2 . (1.113) 2 r
Again, this metric already has the property that radial null geodesics lie at ±45◦, so only a conformal transformation has to be applied taking infinity at finite affine parameter value. Chapter 1. Classical aspects of black holes 27
Figure 1.12: The Penrose diagram of Minkowski spacetime.
Essentially the same transformation as in Minkowski spacetime can be used π π u = tan ω − < ω < 2 2 π π v = tan η − < η < . (1.114) 2 2 With this transformation the metric becomes 2GM ds2 = (2 cos ω cos η)−2[4 1 − dωdη − r2 cos2 ω cos2 η dΩ2] . (1.115) r
Using the fact that 1 sin(η − ω) r∗ = (v − u) = (1.116) 2 2 cos ω cos η one has 2GM r 2 ds˜2 = Λ2ds2 = 4 1 − dωdη − sin2(η − ω)dΩ2 , (1.117) r r∗ where Λ = 2 cos ω cos η . (1.118)
In this metric the asymptotic flatness of Schwarzschild spacetime is manifest. It approaches the metric of compactified Minkowski spacetime (1.105) as r → ∞, with or without fixing t. This means that i0 and I± can be added as before. All r = constant hypersurfaces meet at i+, including the r = 0 hypersurface, which is singular, so i+ is a singular point. Similarly for i−, so these points cannot be added. Near r = 2GM one can introduce Kruskal-Szekeres type coordinates to pass through the horizon. In this way the Penrose diagram for the maximal Chapter 1. Classical aspects of black holes 28 extended Schwarzschild spacetime can be constructed. This is shown on figure 1.13. Sometimes it is convenient to adjust the function Λ so that r = 0 is a vertical line.
Figure 1.13: The Penrose diagram the maximal extended Schwarzschild spacetime.
1.7.2 Gravitational collapse spacetime
From the two examples discussed above the Penrose diagram of the important spacetime of gravitational collapse can be constructed. This spacetime has much more physical relevance since the static Schwarzschild spacetime is an idealization.
Consider again the case of a spherical symmetric contracting shell of massless particles. This shell can be represented by an incoming light-like line in the Penrose diagram of Minkowski space (figure 1.12). The infalling shell divides the Penrose diagram into two regions, an inner and an outer region. The inner region is the interior of the shell throughout time and repre- sents the initial flat spacetime before the shell passes. Since we are only interested in the inner part here, one could really perform the mental cutting and removing of the outer region of the Penrose diagram since this part has to be modified due to the gravitational field exterior to the mass M.
As already mentioned in section 1.2.1, Birkhoff’s theorem states that the spacetime outside the collapsing shell will be that portion of Schwarzschild spacetime cut off by the surface of the shell. Now the same procedure can be done in the Penrose diagram of Schwarzschild spacetime (figure 1.13), but this time the outer region is of interest and the inner region can be removed.
The inner region of Minkowski spacetime can be matched onto the outer region of Schwarzschild spacetime to form the Penrose diagram of a gravitational collapse spacetime, as illustrated in figure 1.14. This matching procedure must be done so that the radius of the local two sphere represented by the angular coordinates (θ, φ) is continuous. In other words, the mathematical identification of the boundaries of the two regions must respect the continuity of the variable r. Since in both cases r varies smoothly from r = ∞ at I− to r = 0, the identification is Chapter 1. Classical aspects of black holes 29 always possible. One should not worry if the two sides needed to be matched don’t have the same length because we are working with conformal pictures and the appropriate stretching or contracting can be performed without changing the physics of these diagrams. That is to say, this deformation will not disturb the form of the light cones.
The final Penrose diagram of a gravitational collapse spacetime makes explicitely clear what was referred to in section 1.4.1, namely that the horizon already forms in the Minkowski space- time inside the contracting shell. It is readily seen that any light ray or timelike trajectory originating behind the dotted line H on figure 1.14 cannot escape to infinity but only end up at the singularity. This defines H as the horizon. In the outer region, H is identical to the surface H+ considered above, and therefore it is found at r = 2MG. In the inner region, the value of r on the horizon grows from an initial value r = 0 to the value r = 2MG when crossing the shell. Notice that H is given by a line at 45◦, implying that it is a null surface, as seen in section 1.6.
Although this model might seem like an unrealistic idealization, it contains all the necessary features to understand a general gravitational collapse spacetime. In figure 1.15 the Penrose diagram for a general gravitational collapse is shown.
Figure 1.14: The construction of the spacetime of a gravitational collapse to a black hole. Chapter 1. Classical aspects of black holes 30
Figure 1.15: The Penrose diagram of a general gravitational collapse.
1.8 No hair conjecture
In the previous sections we made use of Birkshoff’s theorem, which ensures the uniqueness of the Schwarzschild solution on the outside of a spherical symmetric object. In other words, it states that the part of the spacetime on the outside of an object that remains spherical symmetric is necessarily static. It will appear that this idea of uniqueness of a spacetime solution to the Einstein equations can be extended to more general black hole spacetimes.
1.8.1 Kerr-Newman geometry and uniqueness theorems
In order to make a first step towards generalizing the idea of uniqueness we introduce the Einstein-Maxwell action. This is the generalization of the Einstein-Hilbert action which takes into account the electromagnetic field. It is given by [12]
1 Z S = d4x (−g)1/2[R − F F µν] (1.119) 16πG µν where R is the scalar curvature and Fµν is the electromagnetic tensor. The unusual normaliza- tion of the Maxwell term means that the magnitude of the Coulomb force between two point charges Q1, Q2 at large seperation r in flat space is
G|Q Q | 1 2 (1.120) r2 which implies the use of ’geometrized’ units of charge. The source-free Einstein-Maxwell equa- tions derived from (1.119) are
1 G = 2 F F λ − g F F ρσ (1.121) µν µλ ν 4 µν ρσ µν ∇µF = 0 . (1.122) Chapter 1. Classical aspects of black holes 31
The right hand side of (1.121) is the energy-momentum tensor of the electromagnetic field. The source-free Einstein-Maxwell equations have the spherically symmetric static Reissner- Nordstr¨omsolution which generalizes the Schwarzschild solution
2GM GQ2 dr2 ds2 = 1 − + dt2 − − r2dΩ2 (1.123) r r2 2GM GQ2 1 − r + r2 Q A = dt . (1.124) r Where A is the Maxwell 1-form, i.e. the generalization of the electromagnetic potential to a general manifold, and Q is clearly the electric charge. The time component of the metric changes p 2 2 sign when r equals r± = G(M ± M − Q ). So the Reissner-Nordstr¨omsolution only has a horizon when M ≥ Q. In the other case, the electrostatic repulsion would halt the gravitational collapse before a black hole is formed.
Now Birkhoff’s theorem can be generalized straightforwardly to the Reissner-Nordstr¨omcase, stating that the outside of a spherially symmetric charged object is described by the Reissner- Nordstr¨ommetric. Thus, also in the case of a charged object, spherical symmetry implies time-independence of the metric on the outside of the object.
So far, nothing surprising has happened. Even in Newtonian gravity, the gravitational field on the outside of a spherical symmetric mass distribution is the same as if the entire mass would be located in the centre of the distribution. So in a sense, Birkhoff’s theorem could be seen as a relativistic generalization of this feature of Newtonian gravity. But there is more, much more, going on about the idea of uniqueness in the context of black holes. To be able to formulate the theorems below in their simplest form, i.e. without having to use statements involving topology, a theorem by Hawking is used[13, 14]:
A stationary black hole must have a horizon with spherical topology and it must either be static (zero angular momentum), axially symmetric, or both.
A asymptotically flat spacetime is stationary if and only if there exists a Killing vector field k that is timelike near infinity. This means, outside a possible horizon, k = ∂/∂t, where t is a time coordinate. A stationary spacetime is static at least near infinity if it is also invariant under time-reversal. This requires g0i = 0. An asymptotically flat spacetime is axisymmetric if there exists an axial Killing vector field m that is spacelike near infinity and for which all orbits are closed. Coordinates can be chosen such that m = ∂/∂φ, where φ is a coordinate identified modulo 2π, such that m2/r2 → 1 as r → +∞. Thus, as for k, there is a natural choice of normalization for an axial Killing vector field in an asymptotically flat spacetime.
With Hawking’s theorem, the first uniqueness theorem by Israel [15, 16] reads:
Any static black hole has external fields determined uniquely by its mass M and charge Q. More- over, those external fields are the Schwarzschild solution if Q = 0 and the Reissner-Nordstr¨om solution if Q 6= 0. Chapter 1. Classical aspects of black holes 32
Thus, for a black hole Birkhoff’s theorem also works in the reversed direction, i.e. any static black hole has to be spherical symmetric. In general, this is not the case for an object like a star. So black holes really have a special nature compared to ’normal’ objects. Still, Israel’s theorem appears to be only the tip of the iceberg. To proceed, a bigger class of black hole solutions to the Einstein equations needs to be introduced.
The Kerr-Newman three-parameter family in Boyer-Lindquist coordinates, which are the gen- eralization of the Schwarzschild coordinates, is described by the metric
(∆ − a2 sin2 θ) (r2 + a2 − ∆) ds2 = dt2 + 2a sin2 θ dt dφ ρ2 ρ2 (r2 + a2)2 − ∆a2 sin2 θ ρ2 − sin2 θ dφ2 − dr2 − ρ2dθ2 , (1.125) ρ2 ∆ where
ρ2 = r2 + a2 cos2 θ (1.126) ∆ = r2 − 2GMr + a2 + Ge2 (1.127)
The three parameters are M, a and e. The meaning of a is given by
J a = , (1.128) GM with J the total angular momentum of the black hole. For e one has the expression p e = Q2 + P 2 (1.129) where Q is the electric charge and P a hypothetical magnetic monopole charge. The Maxwell 1-form of the Kerr-Newman solution is
Qr(dt − a sin2 θdφ) − P cos θ[adt − (r2 + a2)dφ] A = . (1.130) ρ2
The metric coefficients are independent of t and φ, so the Kerr-Newman family represents a time-independent and axially symmetric solution to the Einstein equations. When a = 0, the Kerr-Newman solution reduces to the Reissner-Nordstr¨omsolution. Hence, with a = 0 and Q = 0 one again finds the Schwarzschild solution. The two-parameter family which follows from (1.125) by putting Q = 0 is called the Kerr solution. Taking φ → −φ effectively changes the sign of a, so one may choose a ≥ 0 without loss of generality. The metric also has the discrete symmetry t → −t , φ → −φ . (1.131)
A Kerr-Newman geometry has a horizon, and therefore describes a black hole, if and only if G2M 2 ≥ G2Q2 + a2. It seems likely that in any collapsing body which violates this contraint, centrifugal forces and/or electrostatic repulsion will halt the collapse before it reaches a size ∼ GM. When G2M 2 = G2Q2 + a2, the solution is called an extreme Kerr-Newman geometry. p 2 2 2 2 2 The horizon is located at r+ = GM + G M − G Q − a ). By looking at the electric and magnetic fields surrounding the Kerr-Newman black hole, one sees that it has an electric charge Chapter 1. Classical aspects of black holes 33
Q and a magnetic dipole moment given by M ≡ Qa. This means that the gyromagnetic radio γ = Q/M, just as for an electron.
In 1970, Carter proved another uniqueness theorem concerning uncharged stationary black holes [17]:
An uncharged, stationary black hole is a member of the two-parameter Kerr family. The param- eters are the mass M and the angular momentum J. In other words, the external gravitational field of the black hole is uniquely determined by its mass and its angular momentum.
The key difference of this theorem with Birkhoff’s theorem is that it only applies to black holes. The Kerr metric is important astrophysically because it is a good approximation to the metric of a rotating star at large distances where all multipole moments except l = 0 and l = 1 are unimportant. The only known solution of Einstein’s equations for which Kerr is exact for µν r > r+ is when T = 0, i.e. the Kerr black hole itself. So it has not been matched to any known non-vacuum solution that could represent the interior of a star, in contrast to the Schwarzschild solution which is guaranteed by Birkhoff’s theorem to be the exact exterior spacetime that matches on to the interior solution for any spherically symmetric star. That there is a thing as Carter’s theorem for black holes but not for stars again emphasises the special nature of black holes.
The importance of these uniqueness theorems must not be underestimated. Since stationar- ity is equivalent with equilibrium, the final state of gravitational collapse is expected to be a stationary spacetime. Carter’s theorem says that if the collapse is to an uncharged black hole then this spacetime is uniquely determined by its mass and angular momentum. This gives extra information about the nature of gravitational collapse to a black hole. Because in con- trast to the case with spherical symmetry, where the geometry outside a gravitational collapse is Schwarschild in character at all stages of the collapse, in the Kerr-Newman case the geomety outside the collapsing body initially departs from Kerr-Newman character. Only well after the collapse occurs, i.e. in the asymptotic future, and in the region at and outside the horizon, is the Kerr-Newman geometry a faithful description of a black hole. Thus, all multipole mo- ments of the gravitational field are radiated away during the collapse to a black hole, except the monopole and dipole moments which can’t be radiated away because the graviton has spin 2.
In other words, the uniqueness theorems come very close to proving that the external grav- itational and electromagnetic fields of a black hole that has settled down to its final state are determined uniquely by the hole’s mass M, charge Q and intrinsic angular momentum J. Thus, a collapse ends with a Kerr-Newman black hole. Other ’quantum numbers’ of the particles that went in the black hole like baryon number, lepton number, strangeness, etc. have no place in the external oberser’s description of a black hole. This is what is meant by the expression that a black hole ’has no hair’.
A natural question to be raised now is, what is so special about mass, angular momentum and electric charge? The answer is that they are all conserved quantities subject to a Gauss type law. Thus, one can determine these properties of a black hole by measurements from afar. Obviously, this reasoning has to be completed by including magnetic monopole charge as Chapter 1. Classical aspects of black holes 34 a fourth parameter because it also is conserved in Einstein-Maxwell theory, it also submmits to a Gauss type law. In the updated no-hair conjecture, the forbidden ’hair’ is any field not of gravitational or electromagnetic nature associated with a black hole.
1.8.2 Classical fields
The above statements of the theorems are all somewhat heuristic. Each theorem and its proof are based on the techniques of global geometry [1,6] and make several highly technical assump- tions about the global properties of spacetime. These assumption seem physically reasonable and innocuous, but they might not be, resulting in a no-hair conjecture rather then a no-hair theorem. Nevertheless, there are several exact results which all support the no-hair conjecture, some of which will be discussed here.
The no-scalar hair theorem can be proven exactly in a rather short and simple manner [18]. Consider the action for a classical, real, massive and minimally coupled (see chapter 2) scalar field 1 Z S = d4x (−g)1/2[gµν∂ ψ∂ ψ + V (ψ2)] , (1.132) 2 µ ν resulting in the field equation µ 2 ∇ ∂µψ − ψV (ψ ) = 0 . (1.133) Because we are looking for a solution in a stationary black hole exterior, assume that the configuration is asymptotically flat and stationary: ∂ψ/∂x0 = 0, where x0 is a time-like variable in the black hole exterior. Multiplying the field equation (1.133) with ψ and integrating over the black hole exterior V at a given x0 gives after integration by parts Z I 3 1/2 ab 2 0 2 µ − d x (−g) [g ∂aψ∂bψ + ψ V (ψ )] + dΣµ ψ∂ ψ , (1.134) V ∂V
0 2 where dΣµ is the 2D element of the boundary hypersurface ∂V and V = ∂V/∂ψ . The indices a and b run over the space coordinates only, so the restricted metric gab is positive definite in the black hole exterior. Now suppose that the boundary ∂V is taken as a large sphere at infinity 2 over all time, which has topology R ⊗ S , together with a surface close to the horizon H, also 2 with topology R ⊗ S . Then so long as ψ decays as 1/r or faster at large distances, which will be true for static solutions of (1.133), infinity’s contribution to the boundary integral vanishes. At the inner boundary, Schwarz’s inequality can be used to state that at every point
µ 2 µ ν 1/2 |ψ∂ ψ dΣµ| ≤ (ψ ∂ ψ∂µdΣ dΣν) . (1.135)
ν As the boundary is pushed to the horizon, a null surface, dΣ dΣν must necessarily tend to 2 µ zero. Thus the inner boundary term will also vanish unless ψ ∂ ψ∂µ blows up at H. But this is unacceptable for a black hole since nothing strange should happen at the horizon if one wants to preserve the equivalence principle. To be more precise, with T µν the energy-momentum tensor µν µ of the field, the physically relevant scalars TµνT and Tµ should remain bounded at H. This µ implies that ∂ ψ∂µψ and V should remain bounded. If V diverges for large arguments, then 2 µ ψ has to remain finite. This implies that ψ ∂ ψ∂µ is bounded on H. But even if V does not diverge at large arguments, so that ψ is allowed to diverge, this will almost certainly cause Chapter 1. Classical aspects of black holes 35
µ ∂ ψ∂µ to diverge. So one can conclude that the boundary term vanishes.
Thus for a generic V if follows that the generalized 4D integral in (1.134) must itself vanish. In 0 2 he case that V (ψ ) is non-negative everywhere and vanishes only at some discrete values ψj, then it is clear that the field must be constant everywhere outside the black hole, taking on one of the values {0, ψj}. The scalar field is thus trivial, either vanishing or taking a constant value as dictated by spontaneous symmetry breaking without the black hole. In particular, the theorem works for the Klein-Gordon field for which V 0(ψ2) = m2, where m is the field’s mass. In that case ψ = 0 outside the black hole. Obviously, this result supports the no-hair conjecture by ruling out that black holes could have parameters associated with a scalar field.
It is remarkable that nowhere is made use of the gravitational field equations. The state- ment that there are no black holes with scalar hair is thus just as true in other metric theories of gravity. Another advantage is that this method can be easily extended to exclude massive vector field hair. A shortcomming is that it does not rule out hair in the form of a Higgs field with a mexican hat potential because then V 0 becomes negative for certain ψ values. But it has been shown by other techniques that a black hole also has no Higgs-hair [19].
When the massive vector field’s mass is removed, gauge invariance sets in and the fundamental
field, which is basically the covariant time component of the vector field Aµ, cannot be required to be bounded at the horizon because it is not gauge invariant [18]. No no-hair theorem can be proven: a black hole can have an electromagnetic field as present in the Kerr-Newman family. By the same logic, one can conclude that the gauge invariance of the non-abelian gauge theories should likewise allow one or more of the gauge field components generated by sources in the black hole to escape from it. Thus gauge fields around a black hole may be possible in every gauge theory. Explicit solutions were found [20, 21]. But these solutions should not be taken into account because they are highly unstable, in fact, almost all known hairy black hole solu- tions in 3+1 general relativity are known to be unstable [22–25]. The only one which is certified to be stable [26], at least in linearized theory, is the Skyrmion hair black hole [27]. It differs from the Schwarzschild one in that it involves a parameter with properties of a topological winding number. This is not an additive quantity among several black holes, so that the Skyrmion black hole may not represent a true exception to the no-hair theorem.
Based upon the no-hair conjecture, it was already known in the 1970’s what the consequences are of the the loss of baryon and lepton number down a black hole. Hartle and Teitelboim [28– 31] have shown that a black hole cannot exert any weak-interaction forces caused by the leptons which have gone down it. A similar analysis to show the absence of strong-interaction forces from baryons that have gone down the hole has been done by Bekenstein and Teitelboim [32– 34]. This implies the non-electromagnetic force between two baryons or leptons resulting from exchange of various force carriers would vanish if one of the particles was allowed to approach a black hole horizon. Chapter 1. Classical aspects of black holes 36
1.8.3 The electric Meissner effect and equilibrium
As mentioned above, an outside observer sees no object ever really arriving at the horizon. So naively, this could lead one to think that when throwing a charged particle in a black hole we will be able to tell forever where the particle entered because we could deduce its position by measuring the electromagnetic field. This appears not to be true. In an article by Cohen and Wald [35] the electrostatic field of a point charge at rest in Schwarzschild space is derived. The solution is then used to study the problem of a point charge being slowly lowered into a nonrotating black hole. It is found that the electric field of the charge remains well behaved as the charge passes the horizon and that all the multipole moments except the monopole fade away. So a Reissner-Nordstrom black hole is produced. This apparant paradox is resolved by the fact that the curvature of spacetime deforms the electric and magnetic fields produced by the charge. In [36] it is shown how there appears to be some sort of ’electric Meissner effect’, bending the field lines of the charge around the black hole as shown on figure 1.16. We will return to this feature in chapter 3 in the context of the membrane paradigm.
Figure 1.16: Electric field lines of a point charge bending around a black hole.
From all this, one can conclude that at the classical level, black holes effectively destroy infor- mation. All information about the initial state of the matter which collapsed to a black hole is lost once the final stationary Kerr-Newman state is reached. This makes the process of black hole formation highly irreversible. The uniqueness of a final stationary state has a strong re- semblance with the concept of thermal equilibrium in statistical physics. In both cases different initial states of the system evolve towards the same stationary final state characterized by a limited amount of macroscopic quantities. This analogy was the basis for the introduction of a notion of entropy for black holes. It had to account for this irreversibility and give meaning to the laws of black hole mechanics which will be discussed in the final section of this chapter.
1.9 Radial null geodesics in black hole spacetimes
For the derivation of the Hawking spectrum in the next chapter it is necessary to go into further detail about radial null geodesics. In particular, a formula relating the constant value of the Chapter 1. Classical aspects of black holes 37 null coordinate u for an outgoing geodesic to the constant value of the null coordinate v for the corresponding ingoing geodesic needs to be derived, and this for geodesics close to the last geodesic that can escape to infinity.
1.9.1 The Schwarzschild black hole
Consider the geodesics in the plane taken at θ = π/2, which can be done without loss of generality. The Schwarzschild metric is independent of t and of φ and, as seen in section 1.1, this implies the existence of two Killing vector fields given by k = ∂/∂t and m = ∂/∂φ and two corresponding conserved quantities
dt 2MG dt E = k · p = g kµpν = g = 1 − (1.136) µν tt dλ r dλ dφ dφ L = m · p = g mµpν = g = r2 , (1.137) µν φφ dλ dλ where (1.20) and (1.21) were used with a rescaling of the conserved quantities such that the mass can be left out of (1.21). Instead of the proper time τ a general affine parameter λ is used along the geodesic.
Requiring the geodesic to be null implies
dxµ dx 0 = µ dλ dλ dxµ dxν = g µν dλ dλ 2MG dr 2 2MG−1 dr 2 dφ2 = 1 − − 1 − − r2 . (1.138) r dλ r dλ dλ
So by means of (1.136) and (1.137) it follows that along a null geodesic
dr 2 L2 2MG E2 = + 1 − . (1.139) dλ r2 r
The radial geodesics are those with L = 0, so
dr = ±E. (1.140) dλ The upper sign corresponds to outgoing geodesics (for r > 2MG), and the lower sign to incoming geodesics. From (1.140) and (1.136) one has
dt 2MG dr 0 = ∓ 1 − dλ r dλ d = (t ∓ r∗) (1.141) dλ where r∗ is again the tortoise coordinate defined by (1.51). This implies again that the null coordinate u = t − r∗ is constant along outgoing radial null geodesics and v = t + r∗ is constant along incoming radial null geodesics. Chapter 1. Classical aspects of black holes 38
Now let C be an incoming radial null geodesic defined by v = v1 for some v1 that passes through the event horizon of the Schwarzschild black hole. Let λ be an affine parameter along this geodesic. The null coordinate u is given along C by some function u(λ). It is the form of this function just outside the event horizon that will determine the spectrum of the particles created by the black hole (see chapter 2). Along the null geodesic C we have
du dt dr∗ = − , (1.142) dλ dλ dλ where dt/dλ is given by (1.136). It follows from (1.140), with the minus sign, that
dr∗ dr∗ dr 2GM −1 = = − 1 − E (1.143) dλ dr dλ r and thus du 2GM −1 = 2 1 − E. (1.144) dλ r Integrating (1.140) along C gives r − 2GM = −Eλ , (1.145) where the integration constant is chosen such that λ is zero at the event horizon. For r > 2GM, the affine parameter λ is negative. With this, we can write
2GM −1 2GM 1 − = 1 − , (1.146) r Eλ and du 4GM = 2E − . (1.147) dλ λ Therefore, along the incoming null geodesic C,
u = 2Eλ − 4GM ln(λ/K1) , (1.148) where K1 is a negative constant. Far from the event horizon, u ≈ 2Eλ, while near the event horizon
u ≈ −4GM ln(λ/K1) . (1.149) The null coordinate u is −∞ at past null infinity I− and +∞ at the event horizon.
Now consider the situation as depicted on figure 1.17. The null ray with constant incoming null coordinate v originates on I−, passes though the center of the collapsing body and becomes the null ray having constant outgoing null coordinate u = u(v). The incoming ray with v = v0 is the last one that passes through the center of the body and reaches I+. Incoming null rays with v > v0 enter the black hole and run into the singularity.
The affine parameter λ along all radially incoming null geodesics that pass through the horizon can chosen such that (1.149) relates u to λ near the event horizon. Then the affine parame- ter distance between the outgoing rays u(v0) and u(v) is constant along the entire length of the geodesics, as measured by the change of the affine parameter λ along any incoming null Chapter 1. Classical aspects of black holes 39
Figure 1.17: Penrose diagram of gravitational collapse with ingoing and outgoing geodesics. ray intersecting the two outgoing null rays. Moving backwards along these outgoing geodesics − through the collapsing body, they become the incoming geodesics that originate on I at v0 and v respectively. The affine separation along the null direction between these two geodesics can be chosen to remain constant along their entire length, as they go from I− to I+. Therefore, − + the affine separation between v and v0 at I is the same as that between u(v) and u(v0) at I . + Because λ = 0 at the horizon, the affine separation between u(v) and u(v0) at I has the value λ that satisfies (1.149) with u having the value u(v).
− Because the coordinate v is itself an affine parameter along I , v − v0 must be related to + the affine separation λ between u(v) and u(v0) on I by
v0 − v = K2λ , (1.150) where K2 is a negative constant. Hence
u(v) = −4MG ln(λ/K1) v − v = −4MG ln 0 K1K2 v − v = −4MG ln 0 , (1.151) K with K a positive constant. This is the relation that will determine the spectrum of the created particles when considering quantum fields in a black hole spacetime in chapter 2.
1.9.2 The Kerr black hole
In this section the Kerr metric, which is the Kerr-Newman metric (1.125) with Q ≡ 0, is con- sidered for GM > a. The Kerr metric is singular at r = r±, the zeros of ∆. In section 1.8.1 it Chapter 1. Classical aspects of black holes 40
√ 2 2 2 was already mentioned that the horizon is located at r = r+ = GM + G M − a . The singu- larities at r = r± are coordinate singularities, they are an insufficiency of the Boyer-Lindquist coordinates. To see this one can introduce null coordinates in analogy to the Schwarzschild case
u = t − r∗ (1.152) v = t + r∗ , (1.153) where the tortoise coordinate is now defined by
dr∗ r2 + a2 = , (1.154) dr ∆ which can be solved explicitely as
∗ r+ r− r = r + 2GM ln|r − r+| − 2GM ln|r − r−| (1.155) r+ − r− r+ − r− GM = r + GM 1 + √ ln|r − r+| G2M 2 − a2 GM −GM 1 − √ ln|r − r−| . (1.156) G2M 2 − a2
But in the Kerr case, also a new angular coordinate χ has to be defined a dχ = dφ − dr . (1.157) ∆ Making the transformation from Boyer-Lindquist to Kerr coordinates (v, r, θ, χ), the analogs of the Eddington-Finkelstein coordinates for a Schwarzschild black hole, the Kerr metric transforms into
(∆ − a2 sin2 θ) (r2 + a2 − ∆) ds2 = dv2 − 2dvdr + 2a sin2 θ dv dχ ρ2 ρ2 (r2 + a2)2 − ∆a2 sin2 θ −2a sin2 θdχdr − sin2 θ dχ2 − ρdθ2 , (1.158) ρ2 which is well-behaved at ∆ = 0.
As mentioned in section 1.8.1, the Kerr metric describes a rotating black hole with angular momentum J = aGM. This can be seen explicitely as follows. Let N± be the hypersurfaces at r = r±. The vector fields normal to N± are given by (1.30):
µr l± = −f±g |N± ∂µ (1.159) 2 2 r± + a ∂ a ∂ = − 2 2 2 f± + 2 2 . (1.160) r± + a cos θ ∂v r± + a ∂χ
They have the property that
2a a2 l2 ∝ g + g + g | = 0 . (1.161) ± vv r2 + a2 vχ (r2 + a2)2 χχ N± Chapter 1. Classical aspects of black holes 41
So N± are null hypersurfaces. It then follows from (1.160) that they are Killing horizons of the Killing vector fields ∂ a ∂ ξ± = + 2 2 , (1.162) ∂v r± + a ∂χ because the metric coefficients are independent of v and χ. Using (1.153), (1.157) and (1.154) this can be written in the original Boyer-Lindquist coordinates as
2 2 ∂ ∂ a ∂ r+ + a ∂ ξ±|N± = + ∗ + 2 2 − ∗ ∂t ∂r r+ + a ∂φ a ∂r ∂ a ∂ = + 2 2 ∂t r± + a ∂φ a = k + 2 2 m . (1.163) r± + a
ν µ As explained in section 1.6 one can find the surface gravities κ± by computing ξ±∇νξ±.A lengthy calculation in appendix A gives
r± − r∓ κ± = 2 2 . (1.164) 2(r± + a )
Thus, we have found that the event horizon r = r+ of a Kerr black hole is a Killing horizon of ξ = k + ΩH m, with a J ΩH = = √ (1.165) 2 2 2 2 4 4 2 r+ + a 2GM(G M + G M − J ) and surface gravity κ = κ+
In coordinates for which k = ∂/∂t and m = ∂/∂φ, the definition of ξ implies
µ ξ ∂µ(φ − ΩH t) = 0 , (1.166) so φ = ΩH t+ constant on orbits of ξ, whereas φ is constant on orbits of k. Note that k is unique. Consider 2 2 (k + αm) = gtt + 2αgtα + α gφφ , (1.167)
2 2 2 2 as long as gtφ is finite and gφφ ∼ r as r → +∞, one has (k + αm) ∼ α r > 0 as r → +∞. So there can be only one Killing vector k that is time-like at infinity and normalized, meaning k2 → +1 as r → +∞.
We can conclude that objects on orbits of ξ rotate with angular velocity ΩH relative to static particles, which are those on orbits of k, and hence relative to stationary observers at infinity. Since the null geodesic generators of the horizon follow orbits of ξ, the black hole is rotating with angular velocity ΩH .
We now return to Boyer-Lindquist coordinates to find the radial null geodesics. Just like in the Schwarzschild case, the two Killing vector fields ∂/∂t and ∂/∂φ imply the existence of two Chapter 1. Classical aspects of black holes 42 conserved quantities
2GMr dt 2aGMr sin2 θ dφ E = 1 − + (1.168) ρ2 dλ ρ2 dλ 2aGMr sin2 θ dt 2a2GMr dφ L = − + r2 + a2 + sin2 θ sin2 θ . (1.169) ρ2 dλ ρ2 dλ
But for the Kerr metric, there is an additional conserved quantity, the Carter constant KC , which is given for null geodesics by [37]
1 dt dφ2 ρ4 dr 2 K = ∆ − (a∆ sin2 θ) − (1.170) C ∆ dλ dλ ∆ dλ dt dφ2 dθ 2 K = a sin θ − (r2 + a2) sin θ + ρ4 (1.171) C dλ dλ dλ
The conserved quantities given by (1.168), (1.169), (1.170) and (1.171), together with the de- mand that the geodesics under consideration are null, leads to the following equations chan- drasekhar
dr 2 ρ4 = [(r2 + a2)E − aL]2 − K ∆ (1.172) dλ C dθ 2 L 2 ρ4 = − aE sin θ − + K (1.173) dλ sin θ C dt 1 ρ2 = {[(r2 + a2)2 − ∆a2 sin2]E − 2aGMrL} (1.174) dλ ∆ dφ 1 L ρ2 = 2aGMrE + (ρ2 − 2GMr) (1.175) dλ ∆ sin2 θ
Radial null geodesics move in planes of constant θ. From (1.173) it is clear that a solution with constant θ = θ0 is only possible if
KC = 0 (1.176) 2 L = aE sin θ0 . (1.177)
With this (1.172), (1.174) and (1.175) become
dr = ±E (1.178) dλ dt r2 + a2 = E (1.179) dλ ∆ dφ aE = . (1.180) dλ ∆ In simplifying (1.174), the identity
(r2 + a2)2 − ∆a2 sin2 θ = ρ2(r2 + a2) + 2a2GMr sin2 θ (1.181) is used. For a = 0, these equations of motion reduce to the ones for radial null geodesics in Schwarzschild spacetime, as found in the previous section. The geodesics specified by (1.178), (1.179) and (1.180) are called the principal null congruence in the Kerr spacetime. A congruence Chapter 1. Classical aspects of black holes 43 is a family of curves such that precisely one curve of the family passes through each spacetime point. As r approaches its horizon value r+ it follows that t → ∞ and φ → ∞. So in Boyer- Lindquist coordinates an incoming object takes an infinite coordinate time to reach the event horizon, but it also winds around the z-axis an infinite amount of times.
The spectrum of the particles created by a Kerr black hole will be determined by the func- tion u(λ), with u the null coordinate as introduced in (1.154) and λ an affine parameter along an incoming geodesic C of the principal null congruence. Along such a geodesic
du dt dr∗ dr = − dλ dλ dr dλ r2 + a2 = 2E , (1.182) ∆ where (1.178), (1.179) and (1.154) were used. (1.178) can be integrated to give
r − r+ = −Eλ , (1.183) where the integration constant was chosen such that λ = 0 at r = r+. The affine parameter λ is negative for r > r+. Combining (1.182) and (1.183) one gets
du (r − Eλ)2 + a2 = 2 + , (1.184) d(Eλ) Eλ[Eλ − (r+ − r−)] which can be solved to give along the incoming null geodesic C 1 Eλ 1 Eλ − (r+ − r−) u = 2Eλ − ln + ln 0 , (1.185) κ+ K1 κ− K1 where κ+ and κ− are given by (1.164). When a → 0, r− → 0 and r+ → 2GM. Then κ+ → 1/4GM and κ− → +∞, so the results of the Schwarzschild spacetime are recovered.
Far outside the event horizon of the Kerr black hole, u ≈ 2Eλ with u approaching −∞ at − I . As r → r+, then u → +∞ with
1 Eλ u ≈ − ln . (1.186) κ+ K1
In the spacetime of a rotating body that undergoes gravitational collapse, consider two outgoing null geodesics C1 and C2 that at late times belong to the principal null congruence. Along these geodesics, u is constant. Let C∞ and C2 intersect the incoming null geodesic C where its affine parameter λ has values λ1 and λ2 respectively. Let these values of λ be negative and of sufficently − small magnitude that (1.186) is valid. The null geodesics C∞ and C2 originate at I as incoming − null geodiscs at v = v1 and v = v2, respectively. As λ1 → 0 , we have v1 → v0, where v0 is the value of v on the last incoming null geodesic that starts from I−, passes through the rotating + collapsing body and reaches I on the outgoing null geodesic u = u(v0). Similarly C2 originates + at a value v2 < v0 and reaches I at u = u(v2). The affine separation along null geodesics − between C∞ and C2 is given by (1.186) with u = u(v2). At I , v is an affine parameter. As in Chapter 1. Classical aspects of black holes 44 the Schwarzschild spacetime, it holds that
v0 − v = K2λ , (1.187) where K2 is a negative constant, and the subscript on v2 is dropped. It follows from (1.186) that 1 v − v u(v) ≈ − ln 0 , (1.188) κ+ K where K is a positive constant. This expression will determine the spectrum of the outgoing particles created by the Kerr black hole.
1.10 Energy extraction in Kerr spacetime
One of the defining properties of a black hole is that nothing, not even light, can escape from behind its horizon. Therefore, it is very surpring that one can nevertheless extract energy out of a rotating black hole. This phenomenon is presented here because it can be seen as some sort of classical analogon to the quantum mechanical process of particle creation by black holes.
1.10.1 The ergosphere of a Kerr black hole
The spacetime of a rotating black hole is asymptotically flat, which means that the metric at spatial infinity is the Minkowski metric. Therefore, the Killing vector field describing time translations for observers at large distances has the simple form
∂ k = . (1.189) ∂t The norm of this vector field is given by
(∆ − a2 sin2 θ) 2GMr k2 = g = = 1 − . (1.190) tt ρ2 r2 + a2 cos2 θ
Where (1.125) was used. So one sees that however kµ is timelike at infinity, it does not have to be timelike everywhere. In particular, it follows that kµ is timelike provided that
r2 + a2 cos2 θ − 2MGr > 0 . (1.191)
For M 2 a2 this implies that p r > GM + G2M 2 − a2 cos2 θ (1.192) √ (or r < GM − G2M 2 − a2 cos2 θ, but this is physically not relevant).
The boundary of this region, i.e. the hypersurface given by p r = GM + G2M 2 − a2 cos2 θ , (1.193) Chapter 1. Classical aspects of black holes 45 is called the ergosphere. The ergosphere intersects the event horizon at θ = 0, π, but it lies outside the horizon for other values of θ. Thus, kµ can become space-like in a region outside the event horizon. This region is called the ergoregion. Because kµ is spacelike, an observer in the ergoregion cannot ’stand still’ relative to a stationary observer at infinity. In fact, an observer in the ergoregion must rotate relative to infinity in the same direction as the black hole. This is an extreme example of the ’dragging of inertial frames’.
Figure 1.18: The ergosphere of a Kerr black hole
1.10.2 The Penrose process
Suppose that a particle approaches a Kerr black hole along a geodesic. If p is its 4-momentum one can identify the constant of motion
E = p · k (1.194) as its energy since E = p0 at infinity. Now suppose that the particle decays into two other particles, one of which falls behind the horizon while the other escapes to infinity.
Figure 1.19: The Penrose process
By conservation of energy one has
E2 = E − E1 . (1.195) Chapter 1. Classical aspects of black holes 46
Normally E1 > 0 so E2 < E, but in this case
E1 = p1 · k (1.196) is not necessarily positive in the ergoregion since k may be space-like there. This means that is is possible for classical particles to have a total negative energy (including rest mass energy) relative to infinity. Thus, if the decay takes place in the ergoregion one may have E2 > E, so energy has been extracted from the black hole.
The event horizon was shown to be a Killing horizon of the Killing vector field ξ = k + ΩH m, so for particles passing through the horizon at r = r+ one has
p · ξ ≥ 0 (1.197) because ξ is future-directed null on the horizon (the horizon is a null surface) and p is future directed time-like or null. It follows that
E − ΩH L ≥ 0 (1.198) where L = −p · m is the component of the particle’s angular momentum in the direction defined by m (only this component is a constant of motion). Thus
E L ≤ . (1.199) ΩH If E is negative, as it is for particle 1 in the Penrose process then L is also negative, so the black hole’s angular momentum is reduced. In the end, one has a black hole of mass M + δM and angular momentum J + δJ, where δM = E and δJ = L, so √ δM 2GM(G2M 2 + G4M 4 − J 2) δJ ≤ = δM , (1.200) ΩH J where (1.165) was used. A little algebra then gives
p δJ 0 ≤ 2G3M 3 + 2GM G4M 4 − J 2 − J δM (1.201) δM and this is equivalent with p δ G2M 2 + G4M 4 − J 2 ≥ 0 . (1.202)
The area of the event horizon is Z √ A ≡ dθ dφ gθθgφφ r=r+ 2 2 = 4π(r+ + a ) p = 8πGM(GM + G2M 2 − a2) (1.203)
This means that energy contraction by the Penrose process is limited by the requirement that δA ≥ 0. In the next section this will be shown to be a special case of the second law of black hole mechanics, which has a striking resemblance with the second law of thermodynamics. Chapter 1. Classical aspects of black holes 47
1.10.3 Superradiance
The Penrose process has a close analogue in the scattering of radiation by a Kerr black hole. For simplicity, consider a massless scalar field ψ. Its energy-momentum tensor is
1 T = ∂ ψ∂ ψ − g (∂ψ)2 . (1.204) µν µ ν 2 µν µ Since ∇µT ν = 0 one has µ ν µν ∇µ(T νk ) = T Dµkν = 0 , (1.205) so one can consider 1 jµ = −T µ kν = −∂µψ k · ∂ψ + kµ(∂ψ)2 (1.206) ν 2 as the future directed (k · J > 0) energy-momentum flux 4-vector of ψ. Now consider the following region S of spacetime, which has the null hypersurface N ⊂ H+ as one boundary.
Figure 1.20: A region of spacetime for superradiance
µ Assume that ∂ψ = 0 at i0. Since ∇µj = 0 one has Z Z 4 1/2 µ µ 0 = d x (−g) ∇µj = dSµ j S ∂S Z Z Z µ µ µ = dSµ j − dSµ j − dSµ j Σ2 Σ1 N Z µ = E2 − E1 − dSµ j (1.207) N where Ei is the energy of the scalar field on the spacelike hypersurface Σi. The energy going through the horizon is therefore Z µ ∆E = E1 − E2 = − dSµ j N Z µ = − dA dv ξµj , (1.208) Chapter 1. Classical aspects of black holes 48 where v is the Kerr time coordinate, defined by (1.153). The energy flux lost per unit of Kerr time is therefore Z Z µ P = − dA ξµj = dA (ξ · ∂ψ)(k · ∇ψ) (1.209) where (1.206) was used, together with the fact that ξ · k = 0 on a Killing horizon N of ξ. This can easily be seen by
2 ξ · k|N = ξ |N −ΩH ξ · m|N
= −ΩH ξ · m|N (1.210) because N is a null surface and therefore ξ, as its Killing vector field, is a null vector on N . Now, N is a fixed point set of m, since m is a Killing vector field (Choose coordinates such that m = ∂/∂φ. The metric (1.125) is independent of φ, so the position of the horizon is independent of φ.) So m must be tangent to N or l · m = 0 where l is normal to N . But ξ ∝ l on N , so
ξ · m|N = 0.
So, explicitely Z ∂ ∂ ∂ψ P = dA ψ + Ω ψ . (1.211) ∂v H ∂φ ∂v For a wave-mode of angular frequency ω
ψ = ψ0 cos(ωv − µφ) , µ ∈ Z (1.212) where µ is the angular momentum quantum number. The time average power lost across the horizon is 1 P = ψ Aω(ω − µΩ) (1.213) 2 0 where A is the area of the horizon. P is positive for most values of ω, but for ω in the range
0 < ω < µΩH (1.214) it is negative, so a wave mode with ω, µ satisfying this inequality is amplified by scattering off the rotating black hole. µ cannot be zero because the amplified field must also take away angular momentum from the black hole. The backreaction on the metric because of the energy loss of the black hole has been neglected in this derivation. Strictly speaking, superradiance is incompatible with stationary black hole spacetimes, but when the superradiant process is sufficiently slow, it is a good approximation.
It is this superradiant phenomenon that made people search for a quantum mechanical process of particle emission by black holes because it has a close resemblance to stimulated emission in atomic physics. And quantum mechanics predicts that where there is stimulated emission, there also is spontaneous emission. That this ’rule’ also applies to the black hole context will become clear in the next chapter. The spontaneous emission will have major consequences for black hole mechanics, which we will discuss in the next section. Chapter 1. Classical aspects of black holes 49
1.11 Black hole mechanics
In the sections about the no-hair conjecture and superradiance, some first signs appeared that there exists an analogy between black holes and thermodynamics. Here, this analogy will be discussed. The remarkable thing is that the results in this section do not refer in any way to the Kerr-Newman family, although they are expected to represent any stationary black hole. The laws of black hole mechanics are derived in a very general way based on some simple physical assumptions about the spacetime concerning causality and asymptotic behaviour.
First of all, the spacetimes considered are supposed to be asymptotically flat at null infin- ity. This means that one can conformally map the spacetime (M, gµν) into another spacetime 0 0 − + (M , gµν) such that the image of M has null boundaries I and I . When using really exotic spacetimes, one should check if these null boundaries satisfy the properties to be found in [11]. But for generic spacetimes, and certainly the ones to be considered in this thesis, these proper- ties are fulfilled automatically.
The second assumption concerns spacetimes containing horizons and states that the part of the spacetime on the outside of the future event horizon should be a regular predictable space- time. The notion of a predictable spacetime will be explained below. First, we note that the assumption forbids the existence of ’naked singularities’, i.e. singularities that are visible to and have influence upon observers at large distances. It is widely believed that no such naked singularity can occur in a physically realistic gravitational collapse. The conjecture that no naked singularities occur is known as the cosmic censorship hypothesis and can be formulated in a relatively precise manner as follows:
Cosmic censorship hypothesis Consider asymptotically flat initial data which are phys- ically achievable on a spacelike hypersurface for a solution of Einstein’s equations with ’suitable’ matter. Then the maximal Cauchy development of these data (i.e. the largest spacetime uniquely determined by these data and Einstein’s equations) is asymptocically flat at null infinity.
The term ’suitable’ appearing in this formulation of the cosmic censorship hypothesis requires some further explanation. Two necessary conditions on matter for it to be ’suitable’ are that it be governed by deterministic (i.e. hyperbolic) differential equations and that it have locally positive energy density. The latter meaning that its energy-momentum tensor T µν satisfies the µ ν dominant energy condition, i.e. Tµνk k ≥ 0 for any future directed time-like vector field k, µ ν and for every future directed causal vector field l (time-like or null), the vector field −T νl is also a future directed causal vector field, implying that matter and energy can not be observed to flow faster than the speed of light. An additional requirement on matter fields for them to be ’suitable’ is that when their differential equations are evolved on a fixed, non-singular and glob- ally hyperbolic spacetime (e.g. Minkowski spacetime), one always obtains globally non-singular solutions. Consequently, any singularities occuring in the Einstein-matter system necessarily would be attributable to gravitational effects.
Now let us come back to the notion of predictability. Let (M, gµν) be an asymptotically flat so- lution of Einstein’s equations with ’suitable’ matter, which contains an asymptotically flat slice Chapter 1. Classical aspects of black holes 50
Σ with compact interior region, and is such that M contains the maximal Cauchy development of data on Σ. Then, by means of the cosmic censorship hypothesis, M will be asymptotically flat at I+, and the domain of dependence D(Σ) of Σ in M will include all events to the future of Σ which are visible from infinity, i.e. it will include I+(Σ) ∩ I−(I+), where I±(A) stands for the future or past development of data on A. This means that phenomena in the region exterior to any black hole that may form is predictable from Σ. It does not follow, even with the above formulation of the cosmic censorship hypothesis, that any events in the black hole need to be contained in D(Σ). However, if the future event horizon H+ is also contained in D(Σ), then the black hole is said to be predictable.
1.11.1 The area theorem
Consider a null surface N and let λ be an affine parameter of the null geodesic generators of N and denote their tangents with respect to this parametrization by kµ. Take α to be such a null geodesic generator and let p ∈ α. The expansion θ of the null geodesic generators of N at µ µ µ a point p is defined by θ = ∇µk . It is the trace of the quantity B ν = ∇νk , which measures geodesic deviation. Now consider an infinitesimal cross-sectional area element of area A of a bundle of geodesics at p and Lie transport this area element along the null geodesic generators of N . A detailed analysis based upon geodesic congruences (see [12]) shows that
dA = θA , (1.215) dλ so θ measures the local rate of change of cross-sectional area as one moves up the geodesics. The geodesic deviation equation governs the rate of change of θ. Using this, and the fact that we are working with generators of a null surface N , one obtains the Raychaudhuri equation [11]
dθ 1 = − θ2 − σ σµν − R kµkν , (1.216) dλ 2 µν µν where σ denotes the shear of the geodesics and Rµν is the Ricci tensor of the spacetime. Now, by means of the Einstein equations with the energy-momentum tensor satisfying the null energy µ ν µ ν condition Tµνk k ≥ 0 it follows that Rµνk k ≥ 0, so
dθ 1 ≤ − θ2 . (1.217) dλ 2 It now immediately follows that 1 1 1 ≥ + λ , (1.218) θ(λ) θ0 2 where θ0 denotes the initial value of θ. Thus, if the geodesics initially are converging, i.e. θ0 < 0, it follows that θ(λ1) = −∞, implying infinite convergence, at some λ1 ≤ 2/|θ0|, provided that the geodesic α can be extended that far. Infinite convergence means that the null geodesic generators form a caustic, which is shown on figure 1.21. The structure of the caustic causes the points p and q on the figure to be contained within the local light cone, so they are time-like separated. Hence, if the family of null geodesic generators is complete, implying that every two points on N are light-like separated, this is a contradiction. Chapter 1. Classical aspects of black holes 51
Figure 1.21: A caustic of null geodesic generators.
With this background, Hawking’s area theorem can be stated and proven [13]:
Area theorem For a predictable black hole satisfying the null energy condition, the area of spatial cross sections of the future event horizon never decreases with time.
It has been shown explicitely for a Schwarzschild black hole in section 1.6, but from its very definition it is clear that a horizon is a null surface since nothing can go faster than the speed of light. If one assumes that the family of null generators of the horizon is complete, then it follows from the reasoning above that θ must be greater than zero at every point on the event horizon. But it appears that the condition of completeness is not strictly necessary for θ to be bigger than zero everywhere on the horizon. For the proof of this, see [2].
Because θ ≥ 0, the cross-sectional area of the horizon locally increases as one moves up the generators of H+. Nevertheless, one has to worry about the possibility that these null geodesic generators might not reach a sufficiently late time slice Σ, e.g. they might terminate on a sin- gularity on the horizon, thus causing the area of H+ ∩ Σ to be smaller than the initial area. However, this possibility cannot occur for a predictable black hole, wherein the event horizon as well as the exterior region is contained in a globally hyperbolic region O of M. Namely, if Σ is a Cauchy surface for this region, then every null geodesic in O must intersect Σ. Thus, if Σ1 + + and Σ2 are Cauchy surfaces with Σ2 ⊂ I (Σ1) , every generator of H at Σ1 must reach Σ2. + + Thus the area of H ∩Σ2 must be at least as large as the area of H ∩Σ1, as was desired to show.
Bekenstein pointed out that there is a close analogy between this result and the second law of thermodynamics in the way that both results assert that a certain quantity never decreases with time, and used it together with thermodynamic considerations to argue that black holes should be assigned an entropy proportional to the area of the event horizon [38]. In the next sec- tion it is shown that other laws of black hole mechanics bear a striking mathematical similarity to the laws of thermodynamics. Chapter 1. Classical aspects of black holes 52
1.11.2 Zeroth law
The area theorem is the only of the classical results which truly concerns the dynamics of black hole event horizons. The zeroth and first laws of black hole mechanics are concerned with equi- librium of quasi-equilibrium processes. That is, they involve stationary black holes, or adiabatic changes from one stationary black hole to another. A key ingredient for the zeroth and first law is given by [1]
Hawking and Ellis Let (M, gµν) be an asymptotically flat spacetime which is stationary. Suppose further that (M, gµν) is a solution of the Einstein equations with matter satisfying suit- able hyperbolic equations, and that the metric and matter fields are analytic. Then the event + horizon H of any black hole in (M, gµν) is a Killing horizon.
Since H+ must be invariant under the Killing vector field k which is time-like at infinity as implied by stationarity, it is obvious that this Killing vector field must be tangent to H+. The above theorem states that if k fails to be normal to H+, then there exists an additional Killing field ξ which is. In the case where k 6= ξ, it can be shown that a linear combination m of these two Killing fields can be chosen so that the orbits of m are closed. Thus, if k 6= ξ, then (M, gµν) is axisymmetric as well as stationary. Note the close relation to the uniqueness theorems of section 1.6. For a stationary black hole, the angular velocity of the horizon ΩH , is defined by ξ = k + ΩH m, just as in section 1.9.2. m is normalized such that closed orbits have periode 2π and k is normalized by requiring k2 → 1 at infinity.
It was mentioned in section 1.6 that κ is constant on a bifurcate Killing horizon. Here, we will make a more general statement:
Zeroth law If Tµν obeys the dominant energy condition then the surface gravity κ is constant on the future event horizon of a stationary black hole.
To see this one uses again Raychaudhuri’s equation (1.216) to show that for a Killing hori- zon N of the Killing vector field ξ it holds that
µ ν Rµνξ ξ |N = 0 . (1.219)
Using this and the fact that ξ2 = 0 on H+ and the above theorem that every horizon of a stationary black hole is a Killing horizon, Einstein’s equations imply
µ ν µ 0 = −Tµνξ ξ |H+ ≡ Jµξ |H+ , (1.220)
µ ν + so that J = (−T νξ )∂µ is tangent to H . It follows that J can be expanded on a basis of tangent vectors to H+ (1) (2) + J = aξ + b1η + b2η on H . (1.221)
2 2 (1) (1) 2 (2) (2) (i) 2 Now J = b1 η · η + b2 η · η since ξ · η = ξ = 0. ξ is the tangent of the generators of the null surface, so it is lies in the time-direction. This implies that η(i) are space-like vectors.
So J is space-like or null (in the case that b1 = b2 = 0). But the dominant energy condition states that it should be time-like or null because of (1.219). So it follows that J ∝ ξ which Chapter 1. Classical aspects of black holes 53 implies 1 ξ J | + = (ξ J − ξ J )| + = 0 . (1.222) [σ ρ] H 2 σ ρ ρ σ H Using the definition of J and again Einstein’s equations one gets
λ 0 = ξ[σTρ] ξλ|H+ λ = ξ[σRρ] ξλ|H+ (1.223)
A lengthy calculation, given in appendix B, then yields
ξ[ρ∂σ]κ|H+ = 0 , (1.224) from which one can conclude that ∂σκ ∝ ξσ. So it follows that
t · ∂κ = 0 (1.225) for any tangent vector t to H+. This shows that κ is constant on H+. It provides a first indication that the surface gravity is an analogue of the temperature. This may seem a weak analogy since there are presumably many constant quantities in a stationary black hole solution. Nonetheless, it is a non-trivial statement. The link between surface gravity and temperature becomes more explicit when considering the first law of black hole mechanics.
1.11.3 First law
For simplicity, consider a vacuum black hole which is altered by dumping in a small amount of matter represented by the energy-momentum tensor ∆Tµν. Then, to first order in ∆Tµν, the change in in the black hole geometry can be neglected when computing the resulting changes in mass and angular momentum of the black hole.
First, we note that the equation ν µ µ ξ ∇νξ = κξ (1.226) is just the geodesic equation in the nonaffinely parametrized form. A Killing parameter v on a µ Killing horizon is defined by ξ ∇µv = 1. Now suppose that (1.226) holds for some parameter τ along the null generators of the Killing horizon , so that ξµ = dxµ/dτ. Then, if one makes the transition to another parameter λ = λ(τ), it follows that
dxν dλ dxν ξν∇ = κ − ξν∂ ln . (1.227) ν dλ ν dτ dλ
So, if we want to maintain the form of (1.226), it should hold that
dλ dλ κ − ξν∂ ln = κ ⇒ ξν∇ = 0 . (1.228) ν dτ ν dτ
ν And because ξ ∇ν is independent of τ, this can be integrated to give
ν ξ ∇νλ = constant . (1.229) Chapter 1. Classical aspects of black holes 54
This allows us to conclude that the geodesic equation of the null generators of a Killing horizon parametrized by a Killing parameter takes the form (1.226).
For the Kerr black hole, v reduces to the previously defined incoming null coordinate, as can be seen by using (1.178), (1.179), (1.180) and (1.154) to rewrite ξ along an outgoing radial null geodesic:
∂ a ∂ ξ = + ∂t r2 + a2 ∂φ ∂ a ∂λ ∂r ∂r∗ ∂ = + ∂t r2 + a2 ∂φ ∂λ ∂r ∂r∗ ∂ ∂ = + , (1.230) ∂t ∂r∗ so v = (1/2)(t + r∗).
Now introduce a new parameter V = eκv , (1.231) which is a generalization of the ingoing Kruskal-Szekeres coordinate. It can be shown that V is an affine parameter along the null geodesics tangent to ξ which generate the horizon. First, use (1.231) to obtain dxµ dxµ dV dxµ ξµ = = = κeκv . (1.232) dv dV dv dV With this one gets dxµ ξν∇ ξµ = ξν∇ κeκv = κξµ . (1.233) ν ν dV Now using the zeroth law, implying that κ is constant, and the fact that v is a Killing parameter, this becomes dxµ dxµ ∇ = 0 , (1.234) dV ν dV from which it follows that V is an affine parameter.
The changes in the mass and angular momentum of the black hole by dumping in a small amount of matter can be written as [2]
Z +∞ Z 2 µ ν ∆M = dV d S ∆Tµνk t (1.235) 0 Z +∞ Z 2 µ ν ∆J = − dV d S ∆Tµνm t , (1.236) 0 where k and m are the same vector fields as before and t is the tangent vector to the null geodesic generators of the horizon with V as an affine parameter. The second integral is over the cross-section S of the horizon corresponding to ’time’ V . Note that the product kµdV is parametrization independent.
On the other hand, the change in area is governed by the Raychaudhuri equation (1.216) ap- 2 µν plied to the exact horizon. To first order in ∆Tµν, the quadratic terms θ and σµνσ can be Chapter 1. Classical aspects of black holes 55 neglected. Hence, by means of the Einstein equations, one obtains
dθ = −8πG∆T tµtν . (1.237) dV µν When integrating the right hand side of this equation over the horizon, the change in the black hole geometry may be neglected. Thus, the tangent vector t on the right hand side can be written in terms of the affine parameter V as
∂ µ tµ = , (1.238) ∂V which by means of (1.231) becomes
1 ∂ µ tµ = . (1.239) κV ∂v
Because v is a Killing parameter it satisfies
∂ ∂ 1 = ξν∇ v = ξν∂ v = + Ω v , (1.240) ν ν ∂t H ∂φ from which follows that 1 1 v = (t + φ) . (1.241) 2 ΩH This allows one to write ∂ 1 ∂ ∂ = + Ω , (1.242) ∂v 2 ∂t H ∂φ which after the proper normalization leads to the identification ∂/∂v = ξ. So (1.239) becomes
1 tµ = ξµ κV 1 = (kµ + Ω mµ) . (1.243) κV H Multiplying both sides of (1.237) by κV and integrating over the horizon, one obtains [39]
Z +∞ Z Z +∞ Z 2 dθ 2 µ µ ν κ dV d SV = −8πG dV d S ∆Tµν(k + ΩH m )t 0 dV 0 = −8πG(∆M − ΩH ∆J) , (1.244) by means of (1.235) and (1.236). The left side of this equation can be evaluated by integration by parts Z Z +∞ Z Z +∞ 2 dθ 2 +∞ d S dV V = d S [θV ]0 − dV θ . (1.245) 0 dV 0 By equation (2.220), the second term on the right is just minus the change in area of the black hole. On the other hand, the first term vanishes since V = 0 at the lower limit and θ must vanish faster than 1/V as V → +∞ is the black hole is to settle down to a stationary final state with finite area. Thus, one obtains κ ∆A = ∆M − Ω ∆J, (1.246) 8πG H Chapter 1. Classical aspects of black holes 56 which is the first law of black hole mechanics.
Now the mathematical analogy between black hole mechanics and thermodynamics is complete. The first law of black hole mechanics has the same form as the first law of thermodynamics
T ∆S = ∆E + P ∆V. (1.247)
Identification of the two laws leads to the conclusion that the black hole mass plays the role of energy in the ordinary first law and −ΩH ∆J is a work term. Because if one considers a rotating body in thermodynamics, one obtains precisely such a −Ω∆J term in the ordinary first law. This leads to the identification of the black hole horizon cross-section area with the entropy
1 S = A, (1.248) 4G implying that κ/2π should take the role of the temperature, an idea which is enfored by the ze- roth law because in ordinary thermodynamics the temperature of a body in thermal equilibrium must be uniform over the body. As mentioned above, the area theorem can be viewed as an analogue of the second law, stating that the entropy cannot decrease. However, it might appear that the nature of the area theorem and the ordinary second law could hardly be more different. The area theorem is a rigorous theorem in differential geometry applicable to predictable black µ ν holes satisfying Rµνk k ≥ 0. The time asymmetry in the area theorem arises from the fact that one is dealing with a future horizon H+ rather than a past horion. On the other hand, the ordinary second law is not believed to be a rigorous law but rather one which holds with over- whelmingly high probability. The time asymmetry of this law arises from a choice of a highly improbable initial state. Nevertheless, there are very few laws of physics which involve time asymmetric behavior of a quantity, so the analogy between these laws should not be dismissed. The third law of ordinary thermodynamics states that it is impossible to achieve absolute zero temperature in a finite series of processes. It appears that also the third law in this formulation has an analogue in black hole physics [16].
Taken all together, the mathematical analogy is very strong. Furthermore, there even is a hint that the analogy may have some physical content as well: the quantity in the laws of black hole physics which plays the role mathematically analogous to the total energy is the mass M of the black hole, which, in general relativity, physically is the total energy of the black hole. However, at this stage, here the physical analogy ends. In classical black hole physics, κ has nothing whatsoever to do with the physical temperature of a black hole, which is absolute zero by any reasonable criterion. We shall see in chapter 2 that this situation changes drastically when a quantum field is placed in the black hole spacetime. Chapter 2
Quantum field theory in curved spacetime
”It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature...” - N. Bohr
Quantum field theory in curved spacetime is a theory wherein gravity is treated in the classical, general relativistic way but matter is treated fully according to the laws of quantum field theory. It is known that the combination of these theories is not an exact description of nature, albeit there is a lot of convincing experimental evidence for both of them seperatly. At the present time many research is going on in finding the true theory for quantum gravity, with string theory and loop quantum gravity as two of the major candidates. But still it can be rewarding to look how quantum fields behave in a curved background. Knowing that this is only an approximation to the real situation, it is necessary to check if one is working within the range of applicability of the approximation. One should at all times stay away from the Planck-scale where quantum gravity dictates the behaviour of matter, space and time. A rule of thumb is that the radius of curvature should be bigger than the Compton wavelength of the field. This is consistent with the fact that there are no problems with putting massless fields on a curved background. In the end, the extension of quantum field theory to a curved background turns out to be richer in consequences than one could have anticipated. It gives rise to the important processes of particle creation in cosmological and black hole spacetimes and it describes inflationary expansion which explains the primordial fluctuations that are now observed in the cosmic microwave background radiation. In this chapter the main focus will be on the way quantum field theory is formulated in a curved background and the implications for black holes.
57 Chapter 2. Quantum field theory in curved spacetime 58
2.1 The formulation of QFT in curved spacetime
Quantum field theory is well-established in Minkowski spacetime, with a consistent mathemat- ical framework, clear physical interpretations and loads of experimental confirmation. In this section it is shown how the necessary concepts can be extended to a general curved spacetime. The first results of quantum field theory in curved spacetime, obtained in the mid-sixties of the past century, were based upon the canonical formulation and contained for example the creation of particles in an expanding universe. Very soon after that it became clear that the loss of Poincar´esymmetry had major impacts on the theory. Many concepts of Minkowski field theory became spoiled and the theory needed a new formulation. This resulted in the algebraic approach which, despite the successes so far, is at the present time still under development.
2.1.1 The canonical approach
First, the pioneering canonical treatment of quantum fields in a curved spacetime is presented. This is done in close analogy to [40].
Consider a curved spacetime with line element
2 µ ν ds = gµν(x)dx dx . (2.1)
The metric will be treated as a given unquantized field. The spacetime is assumed to have a well-defined causal structure and a set of Cauchy hypersurfaces. Let n denote the dimension of the spacetime, with x0 being the time coordinate and x1, x2, ..., xn−1 being the spatial coordi- nates. The matter fields to be quantized are denoted by φa(x).
The action S is constructed from the field φa, so that it is invariant under general coordinate transformations (diffeomorphisms):
0 0 0 0 0 0 0 S[φ (x ), ∇ φ (x ), gµν(x )] = S[φ(x), ∇φ(x), gµν(x)]. (2.2)
The most easy way to do this is to start from the known Minkowski spacetime action and replace partial derivatives by covariant derivatives, the Minkowski metric by a general metric, and introduce the invariant volume element. This is called the minimal coupling description, and is consistent with the equivalence principle:
∂µ → ∇µ
ηµν → gµν n n 1/2 d x → d x |g| , g = det(gµν)
For the Minkowski metric the mostly minus convention (+ − −−) is used. Occasionally, a term which does not vanish at the origin of a locally inertial frame can be added to the Lagrangian to increase the symmetry. Chapter 2. Quantum field theory in curved spacetime 59
The requirement that variations of the action Z n S = d x L(φ, ∇φ, gµν) (2.3) vanish with respect to variations of the fields φa, which are zero on the boundary of integration, yields the equations of motion
∂L ∂L ∂µ − = 0 . (2.4) ∂(∂µφa) ∂φa
The general covariance of the equations of motion is insured by the invariance of the action. Because L is a scalar density it transforms like |g|1/2.
Variation of the action with respect to the field gµν generally does not vanish. However, be- cause of the invariance of S under general coordinate transformations, δS will be zero under the change in gµν induced by an infinitesimal coordinate transformation
xµ → x0µ = xµ − µ(x) , (2.5) where x and x0 refer to the same event in spacetime. Under this transformation the metric transforms like ∂xλ ∂xσ g (x) → g0 (x0) = g (x) . (2.6) µν µν ∂x0µ ∂x0ν λσ And as shown in section 1.1, this leads to the following variation
0 δgµν(x) = gµν(x) − gµν(x) = Lgµν , (2.7) where Lgµν is the Lie derivative of the metric
Lgµν = ∇µν + ∇νµ . (2.8)
µ µ Let’s assume that (x) and ∂λ (x) are zero on the boundary of the region of integration defining the action S (when integrating over an intire infinite spacetime these quantities should drop off to zero sufficiently fast when going to infinity). In that case the variation of the action under the above infinitesimal coordinate transformation becomes Z n δL δS = d x δgµν , (2.9) δgµν(x) because variations in S produced by the changes in the dynamical fields φa vanish as a con- sequence of the equations of motion and the boundary conditions on µ. The invariance of S n 1/2 under coordinate transformations now requires δS to be zero. Hence, with dvx = d x|g| , Z µν δS = − dvx T ∇µν = 0 , (2.10) where the energy-momentum tensor is introduced
δS T µν ≡ −2|g|−1/2 , (2.11) δgµν(x) Chapter 2. Quantum field theory in curved spacetime 60
µν and its symmetry under interchange of indices is used. Because T ν is a vector one gets
µν −1/2 1/2 µν ∇µ(T ν) = |g| ∂µ(|g| T ν) µν µν = (∇µT )ν + T ∇µν . (2.12)
So by integrating and equating the two right hand sides of this expression and using (2.10), one gets Z µν dvx (∇µT )ν = 0 . (2.13)
And because ν is any arbitrary infinitesimal vector the final result is
µν ∇µT = 0 . (2.14)
This expresses the conservation of energy and momentum in a general curved spacetime. The advantage of this construction is that the energy-momentum tensor defined by it is symmetric, which is not true in general for the canonical energy-momentum tensor
µν ∂L µ Θ = ∂νφa − δν L . (2.15) ∂(∂µφa)
µν From δ(g gµν) = 0, it follows that
δ δ g g = − . (2.16) µλ νσ δgλσ δgµν
And with this one can rewrite (2.11) as
δS T = 2|g|−1/2 . (2.17) µν δgµν(x)
µν The sign convention in the definition of T is chosen such that T00 is positive for the classical electromagnetic field with the used mostly minus sign convention for the metric.
One can calculate the symmetric energy-momentum tensor T µν in curved spacetime and then go to the flat spacetime limit, thereby obtaining a symmetric energy-momentum tensor satisfying µν µν ∂µT = 0 in Minkowski spacetime. For any isolated system in Minkowski spacetime, both Θ µν and T yield a unique conserved energy-momentum vector pµ. In curved spacetime it is the symmetric energy-momentum tensor T µν which describes the matter and radiation and couples to the gravitational field through the Einstein field equations.
The Schwinger operator action principle [41] continues to hold in curved spacetime for an arbi- trary infinitesimal transformation of the form
xµ → x0µ = xµ + δxµ φ(x) → φ0(x) = φ(x) + δφ(x) , (2.18) provided that under this transformation δgµν(x) = 0, or
0 gµν(x) = gµν(x) . (2.19) Chapter 2. Quantum field theory in curved spacetime 61
Because in that case the derivation as done in Minkowksi space can be extended to a general spacetime without gµν having to satisfy the Euler-Lagrange equations. The Schwinger operator action principle then gives an expression for the variation of the action:
δS = G(t2) − G(t1) , (2.20) with Z n−1 0 G(t) = d x [πaδφa − Θνδxν] . (2.21)
The integration is on a constant time hypersurface, and
∂L πa = . (2.22) ∂(∂0φa)
G is the generator of the transformation, satisfying
iδF = [F,G] , (2.23) where F is a functional of the φa and πa. Quantization of the theory in the canonical way can be done in complete analogy to Minkowski spacetime, with the relations
0 0 [φa(~x,t), φb(~x , t)] = 0 , [πa(~x,t), πb(~x , t)] = 0 0 0 [φa(~x,t), πb(~x , t)] = iδabδ(~x − ~x ) (2.24) for bosons, and
0 0 {φa(~x,t), φb(~x , t)} = 0 , {πa(~x,t), πb(~x , t)} = 0 0 0 {φa(~x,t), πb(~x , t)} = iδabδ(~x − ~x ) (2.25) for fermions. Here δ(~x − ~x0) is the Dirac delta-function satisfying R dn−1x δ(~x − ~x0)f(~x) = f(~x0) with the integral being performed over the spacelike hypersurface t = constant. One can show that δ(~x − ~x0) and π(~x0, t) transform as spatial scalar densities under transformations of the spatial coordinates on the constant-t hypersurface. Hence, the above commutation and anti- commutation relations are covariant under transformations of the spatial coordinates on the hypersurface, and are therefore the spatially covariant generalization of the corresponding rela- tions that hold in flat spacetime. Also, they are consistent with the equations of motion of the fields, in the sense that if they hold on one constant-t spatial hypersurface, then they also will hold on the other constant-t hypersurfaces.
There are several cases of interest when G is conserved. The simplest is when δxµ = 0 and
δφa is a symmetry of L. Then of course δgµν = 0 and the symmetry of L implies that δS = 0 µ so that G is independent of time. One also has in that case ∂µJ = 0, with
µ ∂L J = δφa . (2.26) ∂(∂µφa)
µ −1/2 µ This follows because J is a vector density, so that we have ∇µ(|g| J ) = 0. Thus, in a curved spacetime the charges and generators of internal symmetries continue to be conserved. Chapter 2. Quantum field theory in curved spacetime 62
The generator G is also conserved when δxµ 6= 0, but is generated by a Killing vector field so that δgµν = 0 holds. Then, invariance of the action under coordinate transformations implies that δS = 0 and it follows from the Schwinger operator action principle that G is constant. For example, if it is possible to choose a coordinate system in which a particular coordinate, say λ λ x , does not appear in gµν, then under a translation in the x direction one has δgµν = 0 and δφ(x) = 0. Then it follows from the definition of G that Z n−1 0 pλ = d x Θλ (2.27)
λµ is constant. Since g and the other components of pµ may not be constants, it does not follow that pλ is constant.
A similar expression involving the symmetric energy-momentum tensor Tµν also holds. Consider again an infinitesimal transformation of the spacetime coordinates, this time slightly rewritten as xµ → x0µ = xµ − ξµ(x) , (2.28) where is a infinitesimal parameter. Now take ξ to be a Killing vector field, i.e. δgµν = Lξgµν = 0. Because of the the conservation of energy in curved spacetime (2.14) one has
µν µν ∇µ(T ξν) = T ∇µξν . (2.29)
Since ξ is a Killing vector field, the Killing vector lemma implies that ∇µξν is anti-symmetric. So because of the symmetry of T µν one gets
µν ∇µ(T ξν) = 0 . (2.30)
µν But because T ξν is a vector, it holds that
µν −1/2 1/2 µν ∇µ(T ξν) = |g| ∂µ(|g| T ξν) , (2.31) and therefore (2.30) becomes 1/2 µν ∂µ(|g| T ξν) = 0 . (2.32) So one gets for the conserved quantity Z o ν Pξ ≡ dvx T ν(x)ξ (x) . (2.33)
In the case where the coordinates are such that gµν is independent of a particular coordinate λ ν ν R 0 x then ξ = δλ is a Killing vector field, and the conserved quantity reduces to Pλ = dvx T λ. In such case, (2.27) should give the same result up to an additive constant independent of the field configuration.
So far, the action, field equations, symmetric energy-momentum tensor, generators of field transformations, commutation or anti-commutation relations and conservation laws were taken under consideration. Up to this point everything was a straightforward extension from flat to curved spacetime. But from this point on, conceptual difficulties will arise and we will be forced to take our conceptions about quantum field theory to a deeper level. The promised richer then Chapter 2. Quantum field theory in curved spacetime 63 expected features of quantum field theory in curved spacetime will begin to reveal themselves in the next section.
2.1.2 A cosmological model
In this section the canonical treatment of a quantum field in an expanding universe is presented [42]. This is done because it gives a first intuitive notion of the difficulties of quantum fields in general spacetimes. So, however this thesis does not concern itself with cosmological purposes, it is still very instructive (albeit in an indirect way) to consider this model in order to develop a deeper understanding of the necessary concepts to be used later in this thesis, such as: the creation of particles by a changing metric, the role of Bogoliubov transformations and the ambiguity of the vacuum state in curved spacetime.
2.1.2.1 The set-up
The spacetime under consideration is described by the spatially flat isotropically changing metric
ds2 = dt2 − a2(t)(dx2 + dy2 + dz2) . (2.34)
The scale factor can have an arbitrary time dependence but the (highly unrealistic) assumption is made that it must asymptotically approach constant values at early and late cosmic time. This cosmic time t is the proper time of a set of clocks on geodesic worldlines that remain at constant values of the spatial coordinates (x, y, z). So, following behaviour is assumed
( a1 as t → −∞ a(t) ∼ (2.35) a2 as t → +∞
Also, a(t) has to be sufficiently smooth and approach the constant values sufficiently fast.
As quantum field, a massless scalar is used according to the rules of the minimal coupling description. Because covariant derivatives are the same as partial derivatives for scalar quanti- ties, the action and Lagrangian density become Z S = dnx L , 1 L = |g|1/2gµν∂ φ∂ φ . (2.36) 2 µ ν This Lagrangian density gives rise to the following field equation
φ = 0 . (2.37) Chapter 2. Quantum field theory in curved spacetime 64
Before continuing the treatment of a scalar field in an expanding universe a scalar product be- tween two spacetime functions, which provides the spacetime with a natural symplectic struc- ture, is introduced: Z n−1 1/2 0ν ∗ ∗ (f1, f2) ≡ i d x |g| g (f1 (~x,t)∂νf2(~x,t) − ∂νf1 (~x,t)f2(~x,t)) Z n−1 1/2 0ν ∗ ←→ ≡ i d x |g| g f1 (~x,t) ∂ νf2(~x,t) (2.38)
Where the integration is taken over a constant t hypersurface. If f1 and f2 are solutions of the field equation which vanish at spatial infinity, then their scalar product is conserved since
d Z ←→ (f , f ) = i dn−1x ∂ (|g|1/2g0νf ∗ ∂ f ) dt 1 2 0 1 ν 2 Z Z n−1 1/2 µν ∗←→ n−1 1/2 iν ∗←→ = i d x |g| ∇µ(g f1 ∂ν f2) − i d x ∂i(|g| g f1 ∂ν f2) = 0 (2.39)
µ −1/2 1/2 µ µ Where the basic identity ∇µV = |g| ∂µ(|g| V ), valid for any vector field V , was used. In terms of a general spacelike hypersurface σ with future-directed unit normal nµ the scalar product is Z 1/2 ν ∗←→ (f1, f2) = i dσ |g| n f1 ∂ν f2 . (2.40) σ This scalar product is conserved under deformations of σ. Suppose σ → σ0 such that σ and σ0 form the spacelike bounderies of a volume v (there may also be timelike boundaries of v at spatial infinity). Then by the Gauss divergence theorem Z n µ 1/2 ∗←→ (f1, f2)σ0 − (f1, f2)σ = i d x ∂ (|g| f1 ∂µ f2) v Z n 1/2 µ ∗←→ = i d x |g| ∇ (f1 ∇ µf2) v = 0 (2.41) as a consequence of the field equation.
Now all the necessary ingredients are acquired to start the first discussion of a quantum field in a specific non-Minkowski spacetime. First, the field equation can be written down explicitely as
−3 3 −2 X 2 a ∂t(a ∂tφ) − a ∂i φ = 0 . (2.42) i
It is convenient to impose periodic boundary conditions in a cube having sides of coordinate length L. Just as in Minkowski spacetime, this is a mathematical trick, with L taken to infinity after physical quantities are calculated. The field operator can now be expandend in the form X φ = {A f (x) + A† f ∗(x)} , (2.43) ~k ~k ~k ~k ~k Chapter 2. Quantum field theory in curved spacetime 65 where i ~ 2πn f = V −1/2eik.~xψ (τ) , ki = (ni ∈ , k = |~k|) (2.44) ~k k L Z And a τ is defined by Z t τ = a−3(t0)dt0 . (2.45)
It follows from the field equation that
d2ψ k + k2a4ψ = 0 . (2.46) dτ 2 k
Now the initial condition is imposed that in the flat spacetime with a = a1, which is approached at early times , we have the Minkowski spacetime field expansion for ψ, but with the constant scale factor a1 taken into account. This implies that for t → −∞, we get the asymptotic behaviour 3 −1/2 −1/2 ~ f~k ∼ (V a1) (2ω1k) exp[i(k.~x − ω1kt)] (2.47) i 0i with ω1k = k/a1. In the spacetime at early times the coordinates can be rescaled like x → x = i 0i a1x , so that we have the usual Minkowski metric and x is the physical or measured distance. 0i i The appropriate rescaled physical momentum is then k = k /a1, and the physical energy of a ~ 0 particle is |k | = k/a1 = ω1k.
From (2.44) it can be seen that the above asympotic behaviour for f~k comes down to putting following asymptotic condition on ψk as t → −∞
3 −1/2 3 ψk(τ) ∼ (2a1ω1k) exp(−iω1ka1τ) , (2.48)
3 where (2.45) was used to replace t by a1τ+ constant, and the constant phase factor was ab- sorbed into the definition of A~k (or into the choise of the time origin).
With (2.47), the scalar product (2.38) becomes
(f , f ) = δ , (f , f ∗ ) = 0 . (2.49) ~k ~k0 ~k,~k0 ~k ~k0
And because the scalar product is conserved, these relations must hold at all times. Similarly, the Minkowski spacetime quantization in the initial flat spacetime implies that the operators
A~k satisfy [A ,A† ] = δ , [A ,A† ] = 0 , (2.50) ~k ~k0 ~k,~k0 ~k ~k0 ~ and that A~k annihilates particles with momentum k/a1 and energy ω1k in the initial Minkowski spacetime. The operators A~k are time-independent so (2.50) is valid at all times.
From (2.51) and (2.50), it follows that the canonical commutation relations hold,
[φ(~x,t), φ(~x0, t)] = 0 , [π(~x,t), π(~x0, t)] = 0 [φ(~x,t), π(~x0, t)] = iδ(~x − ~x0) (2.51) where π is given by (2.22) as 3 π = a ∂tφ = ∂τ φ . (2.52) Chapter 2. Quantum field theory in curved spacetime 66
The proof for the canonical commutation relations (2.51) is as follows. From the field operator expansion and (2.50) one finds X [φ(~x,t), π(~x0, t)] = a3(t) {f (~x,t)∂ f ∗(~x0, t) − f ∗(~x,t)∂ f (~x0, t)} . (2.53) ~k t ~k ~k t ~k ~k
An arbitrary solution of h = 0 can be expanded in the form X h(~x,t) = {f (~k, t)(f , h) − f ∗(~k, t)(f ∗, h)} ~k ~k ~k ~k ~k Z X = −i d3x0 a3(t) {f (~x,t)∂ f ∗(~x0, t) − f ∗(~x,t)∂ f (~x0, t)}h(~x0, t) ~k t ~k ~k t ~k ~k Z X +i d3x0 {f (~x,t)f ∗(~x0, t) − f ∗(~x,t)f (~x0, t)}∂ h(~x0, t) , (2.54) ~k ~k ~k ~k t ~k where the a3(t) comes from the |g|1/2 in the scalar product (2.38). From the above identity one can conclude X a3(t) {f (~x,t)∂ f ∗(~x0, t) − f ∗(~x,t)∂ f } = iδ(~x − ~x0) , (2.55) ~k t ~k ~k t ~k ~k X {f (~x,t)f ∗(~x0, t) − f ∗(~x,t)f (~x0, t)} = 0 . (2.56) ~k ~k ~k ~k ~k
The canonical commutation relation of φ with π follows from (2.55), and that of φ with φ from (2.56).
The equal time commutator of π with π is found by taking the time derivative of the pre- vious expansion of h and recalling that the scalar products are conserved. Then X ∂ h(~x,t) = {∂ f (~k, t)(f , h) − ∂ f ∗(~k, t)(f ∗, h)} , (2.57) t t ~k ~k t ~k ~k ~k where the scalar products can be expanded as before, and using (2.55) one obtains
X { ∂ f ∗(~x,t) ∂ f (~x0, t) − ∂ f ∗(~x0, t) ∂ f (~x,t)} = 0 , (2.58) t ~k t ~k t ~k t ~k ~k from which [π(~x,t), π(~x0, t)] = 0 follows.
For the results of this model both asymptotically flat regions were not required. It would be sufficient, for example, to suppose that at some time in the distant future the universe be- comes asymptotically flat, although it may have never been flat at earlier times. Then the canonical commutation relations would have to hold when a(t) is changing rapidly if they are to hold the far future. Causality then implies that the canonical commutation relations must hold even if the universe never becomes asymptotically flat. Thus, one sees that the canonical commutation relations hold in this curved spacetime with any a(t) as a consequence of their holding in Minkowski spacetime. Chapter 2. Quantum field theory in curved spacetime 67
2.1.2.2 Particle creation
Because of the asymptotic behaviour at early times, in the initial Minkowski spacetime, f~k is a positive frequency solution of the field equation (2.37) and A~k is a particle annihilator. Suppose now that the state vector describing the system in the Heisenberg picture is such that no particles are present at early times. Denoting this state vector by |0i, this implies
~ A~k|0i = 0 , ∀k . (2.59)
The time evolution of ψk(τ) is governed by the ordinary second-order differential equation (2.46). (±) This equation has two linearly independent solutions ψk (τ) with asymptotic behaviour at late times (t → +∞) (±) 3 −1/2 3 ψk ∼ (2a2ω2k) exp(∓iω2ka2τ) , (2.60) where ω2k ≡ k/a2. Therefore, the solution of the differential equation (2.46) can be written (+) (−) written in its most general form by ψk(τ) = αkψk (τ) + βkψk (τ) where αk and βk are two complex contants depending on the form of a(t). So the solution of interest here, with the early time behaviour as imposed by (2.48), must have late time behaviour as t → +∞
3 3 3 −1/2 −ia2ω2kτ ia2ω2kτ ψk(τ) ∼ (2a2ω2k) [αke + βke ] (2.61)
The Wronskian of the differential equation for ψk (2.46) gives the conserved quantity
∗ ∗ ψk∂τ ψk − ψk∂τ ψk = i (2.62) where the right hand side is determined by the imposed early time asymptotic form of ψk (2.48). Filling in the late time asymptotic form of ψk then requires that
2 2 |αk| − |βk| = 1 . (2.63)
From (2.44) and the late time asymptotic form of ψk (2.61), one finds the following late time behaviour for f~k ~ 3 −1/2 −1/2 ik.~x −iω2kt iω2kt f~k ∼ (V a2) (2ω2k) e [αke + βke ] , (2.64) 3 where it is used that a2τ ∼ t + constant at late t and the constant phase factors are absorbed in αk and βk. At this point, the asymptotic form at late times of φ can be written down by regrouping the early time expansion (2.43) according to late time positive and negative frequency parts: X φ(x) = {a g (x) + a† g∗(x)} , (2.65) ~k ~k ~k ~k ~k with g~k being a solution of the field equation which is positive the positive frequency part at late times, 3 −1/2 −1/2 ~ g~k(x) ∼ (V a2) (2ω2k) exp[i(k.~x − ω2kt)] (2.66) and with ∗ † a ≡ αkA + β A . (2.67) ~k ~k ~k −~k Chapter 2. Quantum field theory in curved spacetime 68
~ The a~k can be interpreted as annihilation operators for particles of momentum k/a2 at late times. This interpretation is consistent, since we have
† 2 2 [a , a ] = δ 0 (|αk| − |βk| ) = δ 0 (2.68) ~k ~k0 ~k,~k ~k,~k The transformation of annihilation and creation operators such as that in (2.67) are known as Bogoliubov transformations.
And now happens the final ’magic’ of this model. Using the a~k and the state vector |0i one can calculate the expectation value of the number of particles present at late times in mode ~k:
† 2 hN it→+∞ = h0|a a |0i = |βk| . (2.69) ~k ~k ~k And on the other hand, at early times
† hN it→−∞ = h0|A A |0i = 0 . (2.70) ~k ~k ~k
2 Thus, if a(t) is such that |βk| is non-zero, as is generally the case, particles are created by the changing scale factor of the universe. The above results can readily be extended to the massive case. All results remain the same, but now with the particle energies at early times given by p 2 2 p 2 2 ω1k = (k/a1) + m and at late times by ω2k = (k/a2) + m .
2.1.2.3 Conclusions
In this model the important and very suble role of boundary conditions (at Cauchy surfaces in spacetime) appeared for the first time. It is one of the key features of quantum field theory in curved spacetime because of the absence of the uniformity of spacetime. It will also play a vital role in the derivation of particle creation by black holes.
Particles are created, rather than annihilated, regardless of the relation between a1 and a2. This occurs, despite the time reversal invariance of the field equation, because we have chosen the state vector such that no particles are present at early times. In the time-reversed situation, in which particles are annihilated so that none are present at late times, we would have to take the state vector to be one in which initially there are correlated pairs of particles present. Such an initial state unnatural in a physical context because of the correlations required.
During a rapid change of a(t) in which particle creation is occurring, the particle number is not operationally well defined. Suppose one tries to measure the particle number in a comoving volume (one bounded by geodesics of the spacetime), and that the measurement process takes place in a time interval ∆t. If ∆t is very small, a significant number of particles will be created by the measurement process because of the time-energy uncertainty relation. But if ∆t is large, then a significant number of particles will be created by the change of a(t) during the time of the measurement. There is no value of ∆t for which the minimum uncertainty in the measured particle number is 0. This irreducible imprecision in the measured particle number will become large during a process of rapid particle creation. The uncertainty is reflected in the theory by Chapter 2. Quantum field theory in curved spacetime 69 the absence of an unambiguous or unique definition of a positive frequency solution correspond- ing to physical particles during a period when a(t) is changing. This ambiguity of the particle interpretation of quantum field theory naturally carries over to more general non-static curved spacetimes as well as to spacetimes with event horizons.
The lack of a unique particle interpretation means that in a general curved spacetime, in con- trast to Minkowski spacetime, there is no physically unambiguous unique Heisenberg state vector which can be identified as the vacuum state. This is explicitely the case for the cosmological model above, where although there were no non-gravitational interactions present, the state vector containing no particles at early times was different from the state vector containing no particles at late times. In the context of this model it is also shown that the early time and late time vacuum states are orthogonal to one another, thus giving unitary inequivalent representa- tions of the commutation relations in curved spacetime [43].
The very intimite relation between spin and statistics appears very naturally when putting a quantum field in a general spacetime background. For the scalar field as treated above only Bose-Einstein statistics seems to be consistent with the dynamics of the field [40]. Otherwise, the particles at early times would obey different statistics than the particles at late times, clearly something which is physically not acceptable. This curved-spacetime derivation of the spin-statistics theorem has been extended to higher spin fields [44] and to ghost fields [45].
2.1.3 The loss of Poincar´esymmetry
The above canonical treatment of quantum field theory in curved spacetime exposed some con- ceptual difficulties, like the non-unique definition of annihilation operators, creation operators and the vacuum state. In this section the aim is to take a closer look at exactly what dif- ficulties come up and why they come up. This is done to point out how subtle and highly non-straightforward it is to place a quantum field in a curved background. Because as shown below, quantum field theory as usually formulated contains many elements that are very special to Minkowski spacetime.
It is relatively simple to generalize classical field theory from flat to curved spacetime. That is because there is a clean separation between the field equations and the solutions. The field equations can be easily generalized to curved spacetime in an entirely local and covariant man- ner.
In quantum field theory, ’states’ are the analogs of ’solutions’ in classical field theory. However, properties of these states are deeply embedded in the usual formulations of quantum field theory in Minkowski spacetime. One particular and important example is the Poincar´einvariance of the vacuum state. Chapter 2. Quantum field theory in curved spacetime 70
2.1.3.1 The particle content of the Klein-Gordon field
A simple and concrete example that illustrates the key features is given in [46, 47]. Consider a free, real Klein-Gordon field ψ in flat spacetime
(∂2 − m2)ψ = 0 . (2.71)
The usual route towards formulating a quantum theory of ψ is to decompose it into a series of modes, and then treat each mode by the rules of ordinary quantum mechanics. The field is put in a cubic box of side L with periodic boundary conditions. The field can then be decomposed as a Fourier series in terms of the modes
1 Z ~ ψ ≡ d3x e−ik.~xψ(t, ~x) , ~k = (2π/L)(n , n , n ) . (2.72) ~k L3/2 1 2 3
The Hamiltonian of the system is given by
X 1 H = |ψ˙ |2 + ω2|ψ |2 , ω2 = |~k|2 + m2 . (2.73) 2 ~k k ~k k ~k
So it follows that the free Klein-Gordon field in flat spacetime is equivalent to an infinite collection of decoupled harmonic oscillators. Going to normal modes and quantizing the field by means of the usual commutation relations then gives
1 X 1 ~ ~ ψ(t, ~x) = eik.~x−iωkta + e−ik.~x+iωkta† . (2.74) 3/2 ~k ~ L 2ωk k ~k
States of the free Klein-Gordon field are given the following interpretation: the state denoted by |0i in which all of the oscillators comprimising the Klein-Gordon field are in their ground state is interpreted as ’the vacuum’. States of the form (a†)n|0i are interpreted as ones containing n ’particles’. In an interacting theory, the evolution of the field may be such that it behaves like a free field at early and at late times. In that case, one has a particle interpretation at those early and late times. The relationship between the early and late time particle description of a state is given by the S-matrix and contains a great deal of information about the interacting theory.
The cornerstone of the definition and interpretation of the ’vacuum’ and ’particles’ in the discus- sion above is the ability to decompose the field into its positive and negative frequency parts as can be seen in (2.74). The ability to define this decomposition makes crucial use of the presence of a time translation symmetry in the background Minkowski spacetime. In a generic curved spacetime without symmetries, there is no natural notion of ’positive frequency solutions’ and, consequently, no natural notion of a ’vacuum state’ or ’particles’.
2.1.3.2 The lack of spacetime symmetries
To examine what properties of Minkowski spacetime are used in an essential way in the usual formulation of quantum field theory, the Wightman axioms [48] are considered because they Chapter 2. Quantum field theory in curved spacetime 71 abstract the key features of quantum field theory in Minkowski spacetime in a mathematically clear way. The Wightman axioms are the following:
1. The states of the theory are unit rays in a Hilbert space H that carries a unitary repre- sentation of the Poincar´egroup.
2. The four-momentum that is defined by the action of the Poincar´egroup on the Hilbert space is positive which means its spectrum is contained within the closed future light cone. (= spectrum condition)
3. There exists a unique Poincar´einvariant state, the ’vacuum’.
4. The quantum fields are operator-valued distributions defined on a dense domain D ⊂ H that is both Poincar´einvariant and invariant under the action of the fields and their adjoints.
5. The fields transform in a covariant manner under the action of Poincar´etransformations.
6. At spacelike separations, the quantum fields either commute or anti-commute.
It is clear that the Wightman axioms rely strongly on Poincar´esymmetry, except for the last one. So only this sixth and last axiom can be readily extended to a general spacetime. Since a generic curved spacetime will not possess any symmetries at all, one can certainly not require Poincar´einvariance/covariance or invariance under any other type of spacetime symmetry. In the following, the implications for axioms 2 and 3, and the perturbation and renormalization prescriptions for a quantum field theory are discussed.
Axiom 2
The energy-momentum tensor Tµν of a classical field in curved spacetime is well defined and it µ µ satisfies local energy-momentum conservation in the sense that ∇ Tµν = 0. If t is a vector field on the spacetime that represents time translations and Σ is a Cauchy surface, one can define the total energy E of the field at ’time’ Σ by Z µ ν E = dΣ Tµνt n . (2.75) Σ Classically, the energy-momentum tensor satisfies the dominant energy condition which means µ ν µ Tµνt n ≥ 0 [1]. Thus, classically, one has E ≥ 0. However, unless t is a Killing vector field, which means that the spacetime would be stationary, E will not be conserved, i.e. independent of the choice of Cauchy surface Σ.
In quantum field theory, it is expected that the energy-momentum operator will be well de- µ fined as an operator-valued distribution (see below), and it is expected to be conserved, ∇ Tµν. However, this definition requires spacetime smearing (see (2.76) below). In Minkowski spacetime one can do ’time smearing’ without changing the value of E, since E is conserved, and there is a unique and well defined notion of total energy. However, in the absence of time translation symmetry, one cannot expect E to be well defined at a sharp moment of time. More impor- tantly, it is well known that Tµν cannot satisfy the dominant energy condition in quantum field Chapter 2. Quantum field theory in curved spacetime 72 theory, even when it holds for the corresponding classical theory, so locally energy densities can be arbitrarily negative [47]. It is nevertheless true in Minkowski spacetime that the total energy is positive for physically reasonable states. However, in a curved spacetime without symmetries there is no reason to expect any ’time smeared’ version of E to be positive.
Furthermore, there are simple examples with time translation symmetry, such as a two-dimensional massless Klein-Gordon field in an S1 ⊗ R background, where E can be computed explicitely and is found to be negative [49]. Or, as another example, in de Sitter spacetime there is no globally timelike Killing field and therefore no global notion of energy that is positive [47]. Thus, it appears hopeless to generalize the spectrum condition to curved spacetime in terms of the positivity of a quantity representing the ’total energy’.
Axiom 3 As already noted above, for a free field in Minkowski spacetime, the notion of ’particles’ and ’vacuum’ is intimately related to the notion of ’positive frequency solutions’ which in turn relies on the existence of a time translation symmetry. These notions of a unique ’vacuum state’ and ’particles’ can be straighforwardly generalized to globally stationary curved spacetimes. How- ever, there is no natural notion of ’positive frequency solutions’ in a general, non-stationary curved spacetime.
Nevertheless for a free field on a general spacetime, a notion of ’vacuum state’ can be defined as follows. A state is said to be quasi-free if all of its n-point functions hψ(x1)...ψ(xn)i can be expressed in terms of its 2-point function by the same formula as holds for the ordinary vacuum state in Minkowski spacetime. A state is said to be Hadamard if the singularity structure of its 2-point function hψ(x1)ψ(x2)i in the coincidence limit x1 → x2 is the natural generalization to curved spacetime of the singularity structure of h0|ψ(x1)ψ(x2)|0i in Minkowski spacetime. Thus, in a general curved spacetime, the notion of a quasi-free Hadamard state provides a notion of a ’vacuum state’, associated to which is a corresponding notion of ’particles’.
The problem is that this notion of a vacuum state is highly non-unique. For spacetimes with a non-compact Cauchy surface, different choises of quasi-free Hadamard states give rise, in gen- eral, to unitarily inequivalent Hilbert space constructions of the theory, so in this case it is not even clear what the correct Hilbert space of states should be. In the absence of symmetries or other special properties of a spacetime, there does not appear to be any preferred choise of quasi-free Hadamard state.
Perturbation and renormalization prescriptions The loss of Poincar´esymmetry also has some major consequences for the perturbation rules and the regularization and renormalization prescriptions of a quantum field theory. To begin with, Wick’s theorem becomes ambiguous because it requires normal ordening which relies on the existence of a preferred vacuum state with respect to which the normal ordening is car- ried out. Furthermore, renormalization prescriptions used to define time-ordered products in Minkowski spacetime make use of momentum-space methods and/or Euclidean methods. The momentum-space methods are based on global Fourier transforms of quantities, but a global Fourier transform is a spoiled concept in curved spacetime. The Euclidean methods are based upon analytic continuation and require the ability to ’Euclideanize’ Minkowski spacetime by the Chapter 2. Quantum field theory in curved spacetime 73 transformation t → it, something which clearly is impossible in a general spacetime. Albeit these difficulties might seem insurmountable, it has been showed that quantum field theories which are renormalizable in Minkowski spacetime are also renormalizable in a geneneral spacetime, by using the algebraic framework [50].
2.1.3.3 The algebraic approach
One could see the quest for a preferred vacuum state in quantum field theory in curved space- time like the quest for a preferred coordinate system in classical general relativity. They appear both to be equally meaningless. In general relatity this is manifestly present by formulating the theory in a geometrical way, wherein one does not have to specify a choice of coordinate system. This inspired people to search a formulation of quantum field theory that did not require to specify a choice of state (or representation) to define the theory. This lead to the algebraic approach to quantum field theory in curved spacetime [2, 46, 47] which states that the fundamental observables in quantum field theory are the local fields themselves. The algebraic approach is intimately related to axiomatic quantum field theory.
The algebraic approach makes use of the observation that the Fourier decomposition of the field (2.74) does not make sense as a definition of ψ as an operator at each point (t, ~x). In essence, the contributions from the modes at large |~k| do not diminish rapidly enough with |~k| for the sum to converge. However, these contributions are rapidly varying in spacetime so if we average the right hand side of (2.74) in an appropriate manner over a spacetime region, the sum will converge. This is mathematically translated by the fact that (2.74) defines ψ as an ’operator valued distribution’, i.e. for any smooth test function f with compact support the quantity Z ψ(f) = d4x f(t, ~x)ψ(t, ~x) (2.76) is well defined by (2.74) if the integration is done prior to the summation. The algebraic approach considers particles to have no fundamental meaning in quantum field theory. It derives most results directly from n-point correlation functions. In calculating these correlation functions or results following from them, crucial use is made from the ’spacetime-smearing’ as just described.
However we have just touched upon the algebraic approach very lightly, it has a very rigor- ous mathematical framework. The completion of this mathematical framework is even today a topic of current research [47]. It should also be obvious that the entire domain of quantum field theory in curved spacetime greatly extends the discussion of this section. But for the purpose of this thesis it is not necessary to go into further detail on these matters.
2.2 The Unruh effect
Surprisingly enough, as a first treatment of quantum field theory in curved spacetime we restrict our attention to Minkowski spacetime. The matter being treated here is nevertheless closely related to particle creation by black holes. Chapter 2. Quantum field theory in curved spacetime 74
Although we saw in the previous section that the choice of the vacuum state is not unique in general, there is a natural vacuum state if the spacetime is static. Then, it is natural to let the positive frequency solutions have a t-dependence of the form e−iωt, where the ω are positive constants interpreted as the energy of the particle with respect to the future-directed Killing vector field ∂/∂t. If the spacetime is globally hyperbolic and static, then this choice of positive frequency modes leads to a well-defined and natural vacuum state that preserves the time translation symmetry. This state is called the static vacuum.
Minkowski spacetime has global time-like Killing vector fields which generate time transla- tions in various inertial frames. The sets of positive frequency modes corresponding to these Killing vectors are the same and are the usual positive frequency modes proportial to e−ik0t with k0 > 0, where t is the time parameter with respect to one of the inertial frames. Thus, all these Killing vector fields define the same vacuum state.
Now, consider the boost Killing vector field
∂ ∂ b = z + t , (2.77) ∂t ∂z where z is one of the spatial coordinates. In the region defined by |t| < z in Minkowski spacetime, b is time-like and future-directed. Hence, this region called the right Rindler wedge is a static spacetime with b being the generator of time translations. Thus, one can define the corresponding static vacuum state. However, this vacuum state is not the same as the state obtained by restricting the usual Minkowski vacuum to this region. This observation is crucial in understanding the Unruh effect, as will be explained in the next subsections.
2.2.1 Rindler spacetime
Minkowski spacetime with the metric
ds2 = dt2 − dx2 − dy2 − dz2 (2.78) is of course a static globally hyperbolic spacetime. It can be devided in four distinct parts:
1) |t| < z: right Rindler wedge, is a static globally hyperbolic spacetime 2) |t| < −z: left Rindler wedge, also a static globally hyperbolic spacetime 3) t > |z|: expanding degenerate Kasner universe, globally hyperbolic but not static 4) t < −|z|: contracting degenerate Kasner universe, globally hyperbolic but also not static
These regions are shown on figure 2.1. The curves with arrows are the integral curves of the boost Killing vector field b = z(∂/∂t) + t(∂/∂z). The direction of increasing U = t − z and that of increasing V = t + z are also indicated.
Minkowski spacetime is invariant under the boost
t → t cosh β + z sinh β (2.79) z → t sinh β + z cosh β (2.80) Chapter 2. Quantum field theory in curved spacetime 75
Figure 2.1: The four parts of Minkowski spacetime. where β is the boost parameter. That these transformations are generated by the Killing vector field b can be seen as follows. The integral curves of b are solutions of the set of coupled first order differential equations
dt = z dλ dz = t , (2.81) dλ with λ an arbitrary parameter along the integral curve. This set of coupled first order equations can be rewritten as a decoupled set of second order equations
d2t = t dλ2 d2z = z . (2.82) dλ2 The most general solution of the first second order differential equation is given by t = a cosh λ+ b sinh λ. Applying the appropriate boundary conditions and taking λ = β results in (2.79). (2.80) is analogous.
The boost invariance of Minkowski spacetime motivates the following coordinate transformation
t = ρ sinh η (2.83) z = ρ cosh η , (2.84) Chapter 2. Quantum field theory in curved spacetime 76 where ρ and η take any real value. Then, the Killing vector field b is
∂ρ ∂ ∂η ∂ ∂ρ ∂ ∂η ∂ b = ρ cosh η + + ρ sinh η + (2.85) ∂t ∂ρ ∂t ∂η ∂z ∂ρ ∂z ∂η ∂ cosh η ∂ ∂ sinh η ∂ = ρ cosh η − sinh η + + ρ sinh η cosh η − (2.86) ∂ρ ρ ∂η ∂ρ ρ ∂η ∂ = , (2.87) ∂η and the metric takes the form
ds2 = ρ2dη2 − dρ2 − dx2 − dy2 , (2.88) which is independent of η as expected. The world lines with fixed values of ρ, x and y are the trajectories of the boost transformation of (2.79) and (2.80). Each world line has a constant proper acceleration given by ρ−1= constant. This can be seen using the general formula for the proper acceleration four-vector on an orbit of a vector field ξ as used in section 1.6 of chapter 1
ν µ µ ξ ∇νξ a = ν . (2.89) ξ ξν
Here, one has ξ = b = ∂/∂η. Using the metric (2.88), one obtains
ν 2 ξ ξν = ρ (2.90) ν µ µ ξ ∇νξ = ∇ηξ µ = Γ ηη 1 = − gµρ∂ g 2 ρ ηη = −ρgµρ . (2.91)
Because (2.88) is diagonal, one gets for the proper acceleration four-vector by combining (2.90), (2.91) and (2.89) 1 aµ = (0, , 0, 0) . (2.92) ρ So the proper acceleration becomes
1 a = p−aµa = . (2.93) µ ρ
The coordinates (η, ρ, x, y) cover only the regions with z2 > t2, i.e. the left and right Rindler wedges, as can readily be seen from (2.84).
The Killing vector field b becomes null on the hypersurfaces t = ±z dividing Minkowski space- time into the four regions. It also clearly is orthogonal to the these hypersurfaces, so they are Killing horizons of b. To give a physical interpretation to these horizons, one can use the coor- dinates (ρ, η) of (2.84). The horizons are given by t2 − z2 = 0, which in the (ρ, η)-coordinates becomes ρ = 0. But from (2.93) it is clear that when ρ → 0, a → ∞. So the Killing horizons at ρ = 0 are called acceleration horizons. Chapter 2. Quantum field theory in curved spacetime 77
To discuss quantum fields in the right Rindler wedge, it is convenient to make a further co- ordinate transformation 1 ρ = eaχ (2.94) a η = aτ , (2.95) or in terms of the original variables t and z
1 t = eaχ sinh aτ (2.96) a 1 z = eaχ cosh aτ , (2.97) a where a is a positive constant. Then, the metric takes the form
ds2 = e2aχ(dτ 2 − dχ2) − dx2 − dy2 . (2.98)
This coordinate system will be useful because the world line with χ = 0 has a constant acceler- ation of a. The coordinates (¯τ, χ¯) for the left Rindler wedge are given by
1 t = eaχ¯ sinh aτ¯ (2.99) a 1 z = − eχ¯ cosh aτ¯ , (2.100) a In the next subsection it will be shown that the usual vacuum state for quantum field theory in Minkowski spacetime restricted to the right Rindler wedge is a thermal state with τ playing the role of time, and similarly for the left Rindler wedge.
2.2.2 Accelerating observers and the thermal bath
The two-dimensional massless scalar field in Minkowski spacetime is problematic because of infrared divergences [51]. Nevertheless, this theory is a very good model for explaining the Unruh effect, and it is not necessary to deal with the infrared divergences for this purpose. It also turns out that the Unruh effect in scalar field theory in higher dimensions can be derived in essentially the same manner as in this model. So it captures all the necessary physics to be used in the next sections of this chapter. The model is presented in analogy to [52].
The massless scalar field in two dimensions ψ(t, zo) satisfies the Klein-Gordon equation
∂2 ∂2 − ψ = 0 . (2.101) ∂t2 ∂z2
This field can be expanded as
Z ∞ dk −ik(t−z) −ik(t+z) † ik(t−z) † ik(t+z) ψ(t, z) = √ b−ke + bke + b−ke + b−ke . (2.102) 0 4πk Chapter 2. Quantum field theory in curved spacetime 78
The annihilation and creation operators satisfy
† 0 [b±k, b±k0 ] = δ(k − k ) , (2.103) with all other commutators vanishing. By using the definitions
U = t − z (2.104) V = t + z , (2.105) one can write
ψ(t, z) = ψ−(U) + ψ+(V ) , (2.106) where Z ∞ † ∗ ψ+(V ) = dk [bkfk(V ) + bkfk (V )] , (2.107) 0 with e−ikV fk(V ) = √ , (2.108) 4πk and similarly for ψ−(U). Since the left and right-moving sectors of the field, i.e. ψ+(V ) and ψ−(U), do not interact with one another, only the left moving sector ψ+(V ) is discussed. Thus, the Unruh effect for the theory consisting only of the left-moving sector will be treated. The
Minkowski vacuum state |0iM is defined by
bk|0iM = 0 , (2.109) for all k.
Using the metric in the right Rindler wedge given by (2.98), one finds a field equation of the same form as (2.101) ∂2 ∂2 − ψ = 0 . (2.110) ∂τ 2 ∂χ2 The solutions to this differential equation can be classified again into left and right-moving modes which depend only on
v = τ + ω (2.111) u = τ − ω , (2.112) respectively. These variables are related to U and V as follows
1 U = t − z = − e−au (2.113) a 1 V = t + z = eav . (2.114) a The Lagrangian density leading to the Klein-Gordon equation is invariant under the coordinate transformation (t, z) → (τ, χ). As a result, going through the quantization procedure, one finds exactly the same theory as in the whole of Minkwoski spacetime with (t, z) replaced by (τ, χ). Chapter 2. Quantum field theory in curved spacetime 79
Thus, one has for 0 < V
Z ∞ h R R† ∗ i ψ+(V ) = dω aω gω(v) + aω gω(v) , (2.115) 0 where e−iωv gω(v) = √ , (2.116) 4πω and where R R† 0 [aω , aω0 ] = δ(ω − ω ) , (2.117) with all other commutators vanishing. Notice that the functions gω(v) are eigenfunctions of the boost generator ∂/∂τ.
The field ψ+(V ) can be expressed in the left Rindler wedge with the condition V < 0 < U, by using the left Rindler coordinates (¯τ, χ¯) of (2.100). Definingv ¯ =τ ¯ − χ¯, one obtains equations R (2.115)-(2.117) with v replaced byv ¯ and with the annihilation and creation operators aω and R† L L† aω replaced by a new set of operators aω and aω . The variablev ¯ is related to V by 1 V = − e−av¯ . (2.118) a
The static vacuum state in the left and right Rindler wedges, the Rindler vacuum state |0iR, is defined by R L aω |0iR = aω|0iR = 0 , (2.119) for all ω.
R R L To understand the Unruh effect, one needs to find the Bogoliubov coefficients αωk, βωk, αωk L and βωk, where Z ∞ dk R −ikV R ikV θ(V )gω(v) = √ (αωke + βωke ) (2.120) 0 4πk Z ∞ dk L −ikV L ikV θ(−V )gω(¯v) = √ (αωke + βωke ) , (2.121) 0 4πk
R ikV where θ(x) is the Heaviside function. To find αωk, one multiplies (2.120) by e /2π with k > 0 and integrates over V . Thus, with (2.116), one finds
√ Z ∞ R dV ikV αωk = 4πk gω(V )e 0 2π r k Z ∞ dV = (aV )−iω/aeikV . (2.122) ω 0 2π Chapter 2. Quantum field theory in curved spacetime 80
Now introduce a cut-off for this integral for large V by letting V → V + i, → 0+. Then, changing the integration path to the positive imaginary axis by putting V = ix/k, one finds
πω/2a −iω/a Z ∞ R ie a dx −iω/a −x αωk = √ x e dx ωk k 0 2π ieπω/2a a−iω/a = √ Γ(1 − iω/a) , (2.123) 2π ωk k where Γ(x) represents the gamma-function.
R ikV −ikV To find the coefficients βωk, one replaces e in (2.122) by e . Then, the appropriate substitution is V = −ix/k. As a result, one obtains
ie−πω/2a a−iω/a βR = − √ Γ(1 − iω/a) . (2.124) ωk 2π ωk k A similar calculation leads to
ieπω/2a aiω/a αL = − √ Γ(1 + iω/a) (2.125) ωk 2π ωk k ie−πω/2a aiω/a βL = √ Γ(1 + iω/a) . (2.126) ωk 2π ωk k So one finds following crucial relations for the derivation of the Unruh effect
L −πω/a R∗ βωk = −e αωk (2.127) R −πω/a L∗ βωk = −e αωk . (2.128)
By substituting these relations in (2.120) and (2.121), one finds that the following functions are linear combinations of positive-frequency modes e−ikV in Minkowski spacetime
−πω/a ∗ Gω(V ) = θ(V )gω(v) + θ(−V )e gω(¯v) (2.129) ¯ −πω/a ∗ Gω(V ) = θ(−V )gω(¯v) + θ(V )e gω(v) . (2.130)
One can show that these functions are purely positive-frequency solutions in Minkowski space- time by an analyticity argument as well: since a positive-frequency solution is analytic in the −1/2 −iω/a lower half plane of the complex V -plane, the solution gω(v) = (4πω) V with V < 0 should be continued to the negative real line avoiding the singularity at V = 0 around a small circle in the lower half plane, thus leading to (4πω)−1/2e−πω/a(−V )−iω/a for V < 0.
Equations (2.129) and (2.130) can be inverted to
−πω/a ¯∗ θ(V )gω(v) ∝ Gω(V ) − e Gω(V ) (2.131) ¯ −πω/a ∗ θ(−V )gω(¯v) ∝ Gω(V ) − e Gω(V ) . (2.132)
By subsituting these equations in
Z ∞ h R R† + L L† + i ψ+(V ) = dω θ(V ){aω gω(v) + aω gω (v)} + θ(−V ){aωgω(¯v) + aω gω (¯v)} , (2.133) 0 Chapter 2. Quantum field theory in curved spacetime 81 one finds that the integrand here is proportional to
R −πω/a L† ¯ L −πω/a R† Gω(V )[aω − e aω ] + Gω(V )[aω − e aω ] + h.c. . (2.134)
Because it was derived that the functions Gω(V ) and G¯ω(V ) are positive-frequency solutions with respect to the usual time translation in Minkowski spacetime, one has
R −πω/a L† (aω − e aω )|0iM = 0 (2.135) L −πω/a R† (aω − e aω )|0iM = 0 . (2.136)
These relations uniquely determine the Minkowski vacuum state |0iM as will be explained below.
To explain how the state |0iM is formally expressed in the Fock space on the Rindler vac- uum state |0iR and to show that the state |0iM is a thermal state when it is probed only in the right (or left) Rindler wedge, one uses the approximation where the Rindler energy levels ω are discrete. The rigorous treatment would be to do the calculation in a box and then let the volume of the box go to infinity. But here, a physical and straightforward version of this procedure will be used, not worrying too much about technical restrictions. To do so, write ωi instead of ω and let R R† L L† [aωi , aωj ] = [aωi , aωj ] = δij , (2.137) with all other commutators vanishing. Using the discrete version of (2.135) and the commutators (2.137), one finds
R† R −2πωi/a L† L −2πωi/a h0M |aωi aωi |0M i = e h0M |aωi aωi |0M i + e . (2.138)
R R† L L† The same relation with aωi and aωi replaced by aωi and aωi , respectively and vice versa, can be found using (2.137). By solving these two relations simultaneously, one finds
R† R L† L h0M |aωi aωi |0M i = h0M |aωi aωi |0M i (2.139) 1 = . (2.140) e2πωi/a − 1 Hence, the expectation value of the Rindler-particle number is that of a Bose-Einstein particle in a thermal bath of temperature T = a/2π. Therefore, a uniformly accelerating oberver in Minkowski spacetime will detect a thermal bath of particles, which is the Unruh effect.
Equation (4.42) can be expressed without discretization. Define
Z ∞ R R af = dω f(ω)aω , (2.141) 0 with Z ∞ dω |f(ω)|2 = 1 . (2.142) 0 Then Z ∞ |f(ω)|2 h0 |aR†aR|0 i = dω . (2.143) M f f M 2πω/a 0 e − 1 Exactly the same formula applies to the left Rindler number operator. Chapter 2. Quantum field theory in curved spacetime 82
2.3 Particle creation by black holes
As mentioned in chapter 1, it is possible to classicaly extract energy out of a black hole (the Penrose process) and to have induced emission in the case of rotating black holes (Superradi- ance). Some experience with quantum mechanics learns that in circumstances where there is induced emission, there also is spontaneous emission. So when the development of quantum field theory in curved spacetime arose in the mid-sixties, people tried to find a quantum mechanical mechanism for this spontaneous emission – i.e. spontaneous particle creation from the vacuum.
First, it should be noted that there is nothing wrong whith using quantum field theory in a black hole background as long as one stays far enough from the singularity. As mentioned at the beginning of this chapter, quantum field theory in a curved spacetime is known to be only an approximation to a better and yet to be found physical theory of quantum gravity, but one that is reliable when avoiding Planck-scale phenomena. In a Schwarzschild spacetime the components of the Riemann curvature tensor are of order
1 R(Horizon) ∼ M 2G2 at the horizon. For a large mass black hole they are typically very small. So, however an event horizon is an intrinsically general relativistic phenomenon, there is no danger in using quantum field theory in that region because there are no violent gravitational effects there.
A first notion of particle creation by black holes was made in [53], where it was pointed out that a Reissner-Nordstr¨omblack hole of sufficiently small mass has an electric field that would create electron-positron pairs through the Heisenberg-Euler-Schwinger process. This process was worked out in complete detail in [54]. Further progress was made on particle creation by rotating black holes by Starobinsky [55] and Unruh [56]. The fact that spontaneous particle creation occurs near rotating black holes did not cause much surprise or excitement. The effect is negligble small for macroscopic black holes such as those that would be produced by the collapse of rotating stars. So, unless tiny black holes were produced in the early universe, the effect is not of astrophysical importance. While it is an interesting phenomenon as a matter of principle, it was not surprising or unexpected in view of the ability to extract energy from a rotating black hole by classical processes.
Unruh did the calculation of particle creation by a rotating black hole in the idealized spacetime representing the stationary final state of the black hole. This spacetime necessarily contains also a ”time-reversed black hole”, i.e. a white hole, although white holes are not expected to occur in nature (something which is undoubtedly closely related to the second law of thermodynam- ics). A white hole is a region of spacetime to which nothing can enter, starting from infinity. So for Unruh to get a result, initial conditions had to be imposed on the white hole horizon, expressing that no particles are emerging from the white hole. In this calculation, a seemingly natural choice of the ”in” vacuum state on the white hole horizon was made. But it was not obvious that this choice was physically correct.
And then, in 1974, Hawking realized in his now classic papers [57, 58] that the difficulty of Chapter 2. Quantum field theory in curved spacetime 83
Unruh’s calculation could be overcome by considering the more physically relevant spacetime describing gravitational collapse to a black hole rather than the idealized spacetime describing a stationary black hole (and white hole). Going through the calculation, he found that the results were significantly altered from the results obtained by Unruh. Remarkably, Hawking found that even for a non-rotating black hole, particle creation occurs and produces a steady flux of particles to infinity at late times. And even more remarkably he found that, for a non-rotating black hole, the spectrum of particles emitted to infinity at late times is precisely thermal, at a temperature T = κ/2π, where κ denotes the surface gravity of the black hole.
The implications of Hawking’s results were enormous. They establish that black holes are perfect black (or actually gray) bodies in the thermodynamic sense at non-zero temperature. This tied in perfectly with the mathematical analogy that had previously been discovered be- tween certain laws of black hole physics and the laws of thermodynamics in chapter 1, giving clear evidence that the similarity of these laws is much more than a mere mathematical analogy.
In the following section Hawking’s original results are given for the Schwarzschild black hole and the rotating Kerr black hole, as derived in [58].
2.3.1 Original derivation of the Hawking radiation
The derivation of the Hawking flux takes place in the spacetime of a gravitational collapse as discussed in Chapter 1. This means that at early times the mass that is later to form the black hole is widely dispersed and of sufficiently low density so that the early part of the spacetime is nearly flat. The thermal flux of particles is caused by the formation of an event horizon if the matter collapses. In the calculation the backreaction of particle creation on the metric is neglected. The flux of particles coming from the black hole will make its mass decrease and Schwarzschildradius shrink. But this process is expected to take place sufficiently slow so that when considering particle creation during an amount of time that is small enough, the metric can be taken time-independent. However, after a very long time, the black hole will become explosive because the surface gravity and temperature increase during the shrinking process. At some point, the surface gravity will be so big that the quantum field description is no longer valid. So there are still many open questions about the end state of black hole evaporation.
The field used to derive the Hawking radiation is a massless Hermitian scalar field, satisfy- ing the generally covariant wave equation ψ = 0, or
−1/2 1/2 µν (−g) ∂µ[(−g) g ∂νψ] = 0 , (2.144) because the determinant of the Schwarzschild metric is negative. The created particles observed at late times are created at a short affine distance from the event horizon. Their spectrum is not affected by the regions, such as that inside the collapsing body, where the metric is not stationary. In the spacetime of a body that collapses to form a Schwarzschild of Kerr black hole, one can write the field in the entire spacetime in the form Z † ∗ ψ = dω (aωfω + aωfω) , (2.145) Chapter 2. Quantum field theory in curved spacetime 84
∗ where the fω and fω are a complete set of solutions of the field equation (2.144), with normal- ization
(fω1 , fω2 ) = δ(ω1 − ω2) , (2.146) with the scalar product is defined as in the previous section. The aω are time-independent operators. Then the canonical commutation relations of the field ψ imply that the aω are † annihilators and aω are creation operators obeying
† [aω1 , aω2 ] = δ(ω1 − ω2) (2.147) † † [aω1 , aω2 ] = [aω1 , aω2 ] = 0 (2.148)
The physical interpretation of the aω depends on the choice of the complete set of solutions fω.
Far outside the collapsing body at early times, the definition of the physical particles that would be detected by inertial observers, or equivalently of positive frequency solutions of the
field equation (2.144) is unambiguous. Let the fω be chosen such that at early times and large distances they form a complete set of incoming positive frequency solutions of energy ω. Their asymptotic form on past null infinity, I−, is
−1/2 −1 fω ∼ ω r exp(−iωv)S(θ, φ) , (2.149) where discrete quantum numbers (l, m) are suppressed, and v = t + r is the incoming null coordinate at I−. The factor ω−1/2 is required by the normalization of the scalar product. In − that case, the operators aω are annihilators of particles on I .
At late times, the situation is different because a black hole event horizon has formed. To define a unique solution of the field equation (2.144) outside the black hole, boundary condi- tions have to be given both on the event horizon and on future null infinity, I+. This feature is not a mathematical detail, but really a cornerstone on which the entire derivation is built. It is again an example of how the role of boundary conditions in quantum field theory in curved spacetime cannot be underestimated.
+ − On I , just as on I , the definition of positive frequency solutions is unambiguous. Let the pω be the solutions of the field equation (2.144) that have zero Cauchy data on the event horizon + ∗ and are asymptotically outgoing and positive frequency at I . Assume that pω and pω form a complete set of solutions on I+, satisfying the normalization condition
(pω1 , pω2 ) = δ(ω1 − ω2) . (2.150)
+ The asymptotic form of pω on I is
−1/2 −1 pω ∼ ω r exp(−iωu)S(θ, φ) , (2.151) where again the quantum numbers (l, m) have been suppressed, and u = t − r is the outgoing + null coordinate at I . A wave packet formed by a superposition of the pω is outgoing and localized at large r at late times. Chapter 2. Quantum field theory in curved spacetime 85
The most general solution of the wave equation will have a part that is incoming at the event horizon at late times. Therefore, another set of solutions qω must be introduced such that a superposition of them at late times is localized near the event horizon and has zero Cauchy + + data on I . The precise form of the qω will not affect observations on I , since those observa- ∗ tions can only depend on the pω. Let the qω and qω form a complete set on the horizon with normalization
(qω1 , qω2 ) = δ(ω1 − ω2) . (2.152)
Since wave packets formed from the pω and the qω are in disjoint regions at late times, their conserved scalar product must vanish:
(qω1 , pω2 ) = 0 . (2.153)
One also has
∗ (qω1 , qω2 ) = 0 ∗ (qω1 , pω2 ) = 0 ∗ (pω1 , pω2 ) = 0 .
The field ψ can now be expanded in the entire spacetime as Z † ∗ † ∗ ψ = dω {bωpω + cωqω + bωpω + cωqω} (2.154)
Again using the canonical commutation relations for the field, gives
† [bω1 , bω2 ] = δ(ω1 − ω2) † [cω1 , cω2 ] = δ(ω1 − ω2) , (2.155)
with all other commutators between bω1 and cω2 and their Hermitian conjugates vanishing.
The derivation is done in the Heisenberg picture, so the state vector is independent of time. Let this state vector, |0i, be chosen to have no particles of the field incoming from I−. Thus, |0i is − annihilated by the aω corresponding to particles incoming from I :
aω|0i = 0 , ∀ ω . (2.156)
As in the cosmological model of the previous section, the spectrum of the created particles is determined by the coefficients of the Bogoliubov transformation relating the annihilation oper- ators at early times to the annihilation and creation operators at late times. It is in that spirit that the steps below are made.
∗ The fω and fω are a complete set for expanding any solution of the field equation, so one can write Z 0 ∗ pω = dω (αωω0 fω0 + βωω0 fω0 ) , (2.157) Chapter 2. Quantum field theory in curved spacetime 86
where αωω0 and βωω0 are complex numbers, independent of the coordinates. From (2.150), (2.153) and (2.154) it follows that
bω = (pω, ψ) . (2.158)
∗ Then, expressing ψ and pω in terms of fω0 and fω0 according to (2.145) and (2.157), one gets Z 0 ∗ ∗ † bω = dω (αωω0 aω0 − βωω0 aω0 ) , (2.159)
∗ ∗ 0 00 where it was used that (fω0 , fω00 ) = −δ(ω −ω ). Furthermore, using the expansion of pω (2.157) it follows that Z 0 ∗ ∗ 0 0 0 0 (pω1 , pω2 ) = dω (αω1ω αω2ω − βω1ω βω2ω ) . (2.160)
The coefficients in the expansion of pω (2.157) can be expressed as
∗ βωω0 = −(fω0 , pω) (2.161)
αωω0 = (fω0 , pω) . (2.162)
Now all the necessary general concepts are introduced. First, the Hawking flux will be calculated explicitely for a Schwarzschild black hole, and then for a rotating Kerr black hole.
2.3.1.1 The Schwarzschild black hole
The aim of this section is to calculate the coefficients αωω0 and βωω0 , from which the spectrum of the created particles will follow, for a non-rotating Schwarzschild black hole. The relevant geodesics were discussed in the first chapter. In figure 2.2 the Penrose diagram of the spacetime of a gravitational collapse is shown.
Figure 2.2: The Penrose diagram for matter collapsing to a Schwarzschild black hole.
A wave packet from superposition of the pω for a range of frequencies near a given value ω can be constructed. The coefficients in the superposition can be chosen so that the outgoing wave packet approaches I+ along a null geodesic characterized by a large constant value of u (i.e. at ∗ late times). The components of this wave packet can expressed in terms of the fω0 and fω0 by Chapter 2. Quantum field theory in curved spacetime 87 means of (2.157). Now imagine this wave packet propagating backward in time. Part of it will be scattered back toward infinity by the curved spacetime, and will reach I− as a superposition of the fω0 with frequencies near the original frequency ω. Another part of the wave packet will pass through the center of the collapsing body (ignoring interaction with the matter of the collapsing body, or assuming that the interaction is negligible at sufficiently high frequencies) − ∗ 0 and reach I as a superposition of the fω0 and fω0 having highly blueshifted values of ω ω. This is because when particles leave the near-event horizon region to escape to infinity, they get heavily redshifted. So a particle being present at large distances and large times, travelling back in time towards the horizon, will then undergo the reverse process and get heavily blueshifted. And because this process takes place in the spacetime of a gravitational collapse, no black hole is present at early times. So when the particle propagates even further back in time, going to I−, no redshift (or a certainly smaller redshift) occurs due to the absence of the black hole.
Therefore, the pω in this latter part of the wave packet can be expressed in terms of the fω0 ∗ 0 0 and fω0 with coefficients αωω0 and βωω0 having ω ω. Furthermore, the relevant values of ω become arbitrarily large at sufficiently late times (i.e. as u → ∞) because in the limiting case, namely a particle originating from the black hole horizon, the redshift is infinite. Thus, the late time spectrum of outgoing particles is determined by the asymptotic form of the coefficients for arbitrarily large ω0. It is here that it appears essential to use a gravitational collapse spacetime in the derivation of the Hawking flux. It is clear that the entire spacetime is important in the process. So an idealized, stationary black hole spacetime, as used by Unruh, could never yield the same results.
To determine these coefficients, one traces the latter part of pω, the one going through the collapsing body, back in time along an outgoing geodesic having a very large value of u. The geodesic passes through the center of the collapsing body just before the event horizon has formed, and emerges as an incoming geodesic characterized by a value of v close to v0 as can be seen on figure 2.2. The value of v at which the packet reaches I− is related to the value of u that it had at I+, by v − v u(v) = −4MG ln 0 , (2.163) K as was derived in section 1.9. Here K is a positive constant characterizing the affine parametriza- tion of the geodesic when it is near I+ and I−.
+ The asymptotic form of pω near I is already given by (2.151). The location of the center of this wave packet formed from pω with a small range of frequencies near the value of ω is determined by the principle of stationary phase. It follows that at early times, the components pω forming the part of the wave packet that passes back through the collapsing body and reaches I− at v have (to within a normalization constant) the form on I−
−1/2 −1 pω ∼ ω r exp(−iωu(v))S(θ, φ) , (2.164) with u(v) given by (2.163) and v < v0, because otherwise the wave packet would end up in the − black hole. The fω0 in the expansion of pω (2.157) have an asymptotic form near I given by − (2.167) with v < v0, because this part of the wave packet cannot reach I at v > v0. Chapter 2. Quantum field theory in curved spacetime 88
Using these early time asymptotic forms for pω and fω0 , one can show with Fourier’s theo- rem that
Z v0 0 1/2 ω iω0v −iωu(v) αωω0 = C dv e e (2.165) −∞ ω Z v0 0 1/2 ω −iω0v −iωu(v) βωω0 = C dv e e , (2.166) −∞ ω where C is a constant. Now substituting (2.163) for u(v) and introducing the new variable s ≡ v0 − v in the expression for αωω0 and s ≡ v − v0 in the expression for βωω0 one gets
∞ 0 1/2 Z ω 0 0 s −iω s iω v0 αωω0 = C ds e e exp[iω4MG ln ] (2.167) 0 ω K 0 0 1/2 Z ω 0 0 s −iω s −iω v0 βωω0 = C ds e e exp[iω4MG ln − ] . (2.168) −∞ ω K
In the equation for αωω0 (2.167), the contour of integration along the real axis from 0 to ∞ can be closed by a a quarter circle at infinity and by the contour along the imaginary axis from −∞ to 0. Because there are no poles in the enclosed quadrant of the complex plane, and the integrand vanishes at infinity, the integral from 0 to ∞ along the real s-axis equals the integral from 0 to −i∞ along the imaginary s-axis.
Similarly, in the expression for βωω0 (2.168), the integral along the real axis in the complex s plane from −∞ to 0 can be joined by a quarter circle at infinity to the contour along the imaginary s-axis from −i∞ to 0, thereby resulting in a closed contour. One gets that the inte- gral from −∞ to 0 equals the integral from −i∞ to 0, for the same reasons as before. Therefore, putting s ≡ is0, it follows
0 0 1/2 0 Z ω 0 0 0 is 0 ω s iω v0 αωω0 = −iC ds e e exp[iω4MG ln ] (2.169) −∞ ω K 0 0 1/2 0 Z ω 0 0 0 −is 0 ω s −iω v0 βωω0 = iC ds e e exp[iω4MG ln ] . (2.170) −∞ ω K
Now the multiple-valued complex logarithm has to be dealt with. One gets a single-valued natural logarithm function by taking the cut in the complex plane along the negative real axis. So for s0 < 0, as in the integrals above, the complex logarithm be written as
ln(is0/K) = ln(−i|s0|/K) = −i(π/2) + ln(|s0|/K) , and ln(−is0/K) = ln(i|s0|/K) = i(π/2) + ln(|s0|/K) .
This is because to get from the negative part of the imaginary axis to the positive part of the real axis, one has to perform a counterclockwise rotation over π/2, and to get from the positive part of the imaginary axis to the positive part of the real axis, a clockwise rotation over π/2 is Chapter 2. Quantum field theory in curved spacetime 89 required.
So (2.169) and (2.170) become
0 0 1/2 0 0 Z ω 0 0 |s | iω v0 2πωMG 0 ω s αωω0 = −iCe e ds e exp[iω4MG ln ] (2.171) −∞ ω K 0 0 1/2 0 0 Z ω 0 0 |s | −iω v0 −2πωMG 0 ω s βωω0 = iCe e ds e exp[iω4MG ln ] . (2.172) −∞ ω K
And this leads to the important result
2 2 |αωω0 | = exp(8πMGω)|βωω0 | , (2.173) for the part of the wave packet that was propagated back in time through the collapsing body just before if formed a black hole.
For the components pω of this part of the wave packet, one has the scalar product,
(pω1 , pω2 ) = Γ(ω1)δ(ω1 − ω2) , (2.174)
+ where Γ(ω1) is the fraction of an outgoing packet of frequency ω1 at I that would propagate backward in time through the collapsing body to I−. One can see this is the following way. Let (2) (1) pω denote the components of this part of the wave packet, and let pω denote the components of the part of the wave packet that if propagated backward in time would be scattered from the spacetime outside the collapsing body and would travel back in time, reaching I− with the same frequency ω as when it had when it started from I+. This is because this latter part of the wave packet stays at all time in the outside region of the black hole (and later the mass that collapsed to form the black hole) so that its blueshift when approaching the mass and its redshift when going away from the mass cancel each other exactly.
(1) (2) − Because pω and pω propagate to disjoint regions on I (i.e. v > v0 and v < v0 respec- (1) (2) tively), they are orthogonal. With pω = pω + pω , it then follows that
(1) (1) (2) (2) (pω1 , pω2 ) = (pω1 , pω2 ) + (pω1 , pω2 ) . (2.175)
So from this and (2.150) one has
(1) (1) (pω1 , pω2 ) = Γ(ω1)δ(ω1 − ω2) (2.176) (2) (2) (pω1 , pω2 ) = (1 − Γ(ω1))δ(ω1 − ω2) (2.177)
+ where Γ(ω1) is the fraction of the packet of frequency ω1 at I that would propagate back through the collapsing body to reach I−
It then follows from (2.160) and (2.175) that Z 0 ∗ ∗ 0 0 0 0 Γ(ω1)δ(ω1 − ω2) = dω (αω1ω αω2ω − βω1ω βω2ω ) , (2.178) Chapter 2. Quantum field theory in curved spacetime 90
(2) where αωω0 and βωω0 now refer to the coefficients in the expansion of pω in terms of the fω0 ∗ and fω0 as in (2.157).
The part of bω in (2.158) that is of interest is
(2) (2) bω = (pω , ψ) . (2.179)
(2) To simplify the notation, from now on bω will refer only to bω .
The information about the particles that are created in the collapse of the body to form a black hole should be contained in bω, but one encouters an infinity by straightforward evalua- tion of Z † 0 2 h0|bωbω|0i = dω |βωω0 | . (2.180)
† This infinity is a consequence of the δ(ω1 − ω2) that appears in (2.178). Since h0|bωbω|0i is the total number of created particles per unit frequency that reach I+ at late times in the wave (2) pω , this total number is infinite (neglecting the change in the mass of the black hole of course) because there is a steady flux of particles reaching I+ at late times.
One way to see this is to replace δ(ω1 − ω2) in (2.178) by
1 Z T/2 i(ω1−ω2)t δ(ω1 − ω2) = lim dt e . (2.181) T →∞ 2π −T/2
Then, for ω1 = ω2 = ω (2.178) can be written as Z 0 2 2 lim Γ(ω)(T/2π) = dω (|αωω0 | − |βωω0 | ) (2.182) T →∞ Z 0 2 = [exp(8πMGω) − 1] dω |βωω0 | , (2.183) where (2.173) was used. Hence,
† −1 h0|bωbω|0i = lim (T/2π)Γ(ω)[exp(8πMGω) − 1] . (2.184) T →∞
The interpretation of this is that at late times, the number of created particles per unit angular frequency and per unit time that passes through a surface r = R, with R much larger than the Schwarzschild radius, is Γ(ω) 1 (2.185) 2π exp(8πMGω) − 1 (Note that the number per unit frequency per unit time has no factor (2π)−1.)
Recall that the quantity Γ(ω) is the fraction of a purely outgoing wave packet that when prop- agated from I+ backward in time would enter the collapsing body just before it had formed a black hole. At sufficiently late times this fraction is the same as the fraction of the wave packet that would enter the black hole past event horizon if the collapsing body were replaced in the spacetime by the analytic extension of the black hole spacetime. This means that Γlm(ω) is also the probability that a purely incoming wave packet that starts from I− at late times will enter Chapter 2. Quantum field theory in curved spacetime 91 the black hole event horizon, that is, will be absorbed by the black hole.
Therefore (2.185) implies that a Schwarzschild black hole emits and absorbs radiation exactly like a gray body of absorptivity Γ(ω) and temperature T given by
1 kT = 8πMG κ = (2.186) 2π where k is Boltzmann’s constant, and κ = 1/4MG is the surface gravity of a Schwarzschild black hole as derived in section 1.6.
2.3.1.2 The Kerr black hole
Calculating the Hawking flux for a rotating Kerr black hole is essentially the same as in the non-rotating case, with two basic changes.
First, the radial geodesics in the Schwarzschild spacetime are replaced by the principal null congruence of geodesics in the Kerr spacetime, as derived in chapter 1. This means that as one traces back in time from I+ to I− the part of an outgoing wave packet that passes through the collapsing body just before the event horizon has formed, the value of u that the wave packet has on I+ is related to the value of v it had on I− by
1 v − v u(v) ≈ − ln 0 , (2.187) κ K as derived in section 1.9.2, where
r+ − r− κ = κ+ = 2 2 (2.188) 2(r+ + a ) is the surface gravity of the Kerr black hole as calculated in appendix B.
The second difference is that the event horizon of the Kerr black hole has angular velocity dφ/dt = ΩH . As one approaches arbitrarily close to the null generators of the event horizon ˜ at r+, both φ and t diverge, but the angular coordinate φ+ = φ − ΩH t is well behaved in the (2) vicinity of r+. In tracing an outgoing wave packet with components pω back in time into the collapsing body just before it has fallen within the event horizon, the angular coordinate φ˜+ is appropriate as the wave packet passes into the collapsing body.
(2) + The result is that if pω has the form exp[−iωu + imφ] at I , then it has the form exp[−i(ω − 0 − 0 mΩH )u(v) + imφ ] at I , where φ is the azimuthal angular coordinate in an inertial coordinate system far outside the collapsing body at early times. m is the azimuthal quantum number, which may have either sign.
As a consequence of these two differences between the non-rotating and rotating cases, the quan- tity ω in the right-hand sides of (2.164) through (2.171) is replaced by the quantity ω − mΩH , Chapter 2. Quantum field theory in curved spacetime 92 and u(v) is replaced by expression (2.187), so 4MG in (2.167)-(2.171) is replaced by κ−1. Hence, for a rotating black hole one finds
2 −1 2 |αωω0 | = exp[2πκ (ω − mΩH )] |βωω0 | (2.189) instead of (2.173).
It then follows, as in the previous section, that the average number of particles created in a wave packet that reaches I+ with energy ω and angular momentum quantum numbers l, m is
−1 −1 hNωlmi = Γlm(ω){exp[2πκ (ω − mΩH )] − 1} , (2.190) where the surface gravity κ is given by (2.188), and Γlm(ω) is the same as the fraction of a similar wave packet incident on a Kerr black hole that would be absorbed by the black hole. Thus, the Kerr black hole acts like a gray body at temperature κ kT = (2.191) 2π where k is again Boltzman’s constant. This is the same equation as for a Schwarzschild black hole, but only the expression for the surface gravity is different.
If (2.190) is to make sense, i.e. hNωlmi > 0, Γlm(ω) has to be negative when ω < mΩH . This means that when an incoming wave packet with ω < mΩH is sent towards a Kerr black hole, the backscattered part of the wave packet returns with a larger amplitude than the original incoming packet. This is the superradiant scattering phenomenon that was discussed in section 1.10.3. Superradiance can be thought of as stimulated pair production caused by the incoming boson.
For fermions one finds the expression
−1 −1 hNωlmiferm = Γlm(ω){exp[2πκ (ω − mΩH )] + 1} . (2.192)
The +-sign now implies that Γlm(ω) remains positive at all frequencies. So there is no radiant scattering for fermions because of the Pauli exclusion principle.
For a charged rotating black hole, it can also be shown [58] that the average number of particles of charge e emitted in mode ω, l, m has the same form of (2.190), but with ω − mΩH − eΦ appearing in the exponential, where Φ is the electrostatic potential of the black hole, and with the expressions for the surface gravity κ and gray body factors Γ appropriate for a rotating charged black hole. The temperature of such a black hole satisfies again equation (2.191) with the appropriate expression for the surface gravity.
2.3.1.3 Final remarks
The above derivation of the Hawking radiation can easily be generalized to the case of non- spherical symmetric gravitational collapse. The late time emission depends only on the final Chapter 2. Quantum field theory in curved spacetime 93 state of the black hole. The detailed nature of the collapse and the manner in which the black hole ’settles down’ to its final state are not relevant. So one can conclude from the uniqueness theorems of section 1.8.1 that we have actually treated the most general case of Hawking radi- ation, at least, with respect to the spacetime in which the emission takes place.
The generalization to physical relevant interacting fields is not so evident. To adress this issue, we mention that the existence of the Hawking flux has also been derived in the algebraic frame- work of quantum field theory in curved spacetime as described in section 2.1.3.3 [59]. There, the derivation of the thermal behavior of the quantum field at asymptotically late times is shown to arise from the singularity structure of the two-point function at arbitrary short distances. However, even ignoring possible new effects arising from the quantum nature of gravity itself at distance scales smaller than the Planck length, it is unreasonable just to assume that the simple linear field model considered in the derivation above will provide an accurate model to a realistic field theory at ultra-short distance scales. Thus, one might question whether the particle creation effect will occur for nonlinear fields even if these fields can be treated as non- interacting on large distance scales or equivalently, at low energies. In response to this issue, it should be noted that the Unruh effect, which has the same physical and mathematical origin as the Hawking effect, is proven to continue to hold for nonlinear fields in Minkowski spacetime by a theorem of Bisognano and Wichmann [60]. Furthermore, there is strong evidence based upon the analytic continuation of propagators to a Euclidean curved spacetime that the Unruh effect even continues to hold for nonlinear fields in static curved spacetimes [2]. So although there is no conclusive proof that the Hawking effect continues to hold for nonlinear fields, all the evidence currently available points to the fact that it does. Together with it’s role in completing black hole thermodynamics (see section 2.5 below) this makes that there is very little doubt about the validity of the Hawking effect for interacting fields.
Although the emission of Hawking radiation has a very low intensity, especially for large black holes, after a sufficient amount of time backreaction effects on the metric will become relevant. By conservation energy it is clear what will happen: the mass of the black hole, and thereby its Schwarzschild radius, will decrease because of the energy that is being emitted under the form of Hawking particles. As the black hole gets smaller, it gets hotter and so starts to radiate faster. As the temperature rises, it exceeds the rest mass of subsequently more and more mas- sive particles. So at first, only photons and neutrino’s will be emitted, then the temperature increases and particles such as electrons and muons would will begin to constitute the Hawking flux until eventually all types of particles will take place in the radiation process. At the time the black hole temperature reaches the strong interaction energy scale, a large amount of energy will be emitted at time scales of 10−23s. So whatever theory dictates the laws of physics at the Planck scale, it is very likely that the evaporation process will end with an explosion, completely erasing the black hole.
After the evaporation process, the energy that was orginally in the black hole will be uni- formly spread throughout space. Because of the low emission rate of the Hawking radiation the energy density will be negligible and the final state of the evaporation process will be flat spacetime. So after an appropriate ’gluing job’ (see section 1.7.2) between the Penrose diagram of a gravitational collapse spacetime and that of Minkowski spacetime one gets the Penrose diagram of a spacetime for gravitational collapse of matter to a black hole and the subsequent Chapter 2. Quantum field theory in curved spacetime 94 evaporation process leading to flat spacetime. This diagram is given on figure 2.3, where B represents the boundary of the collapsing body.
Figure 2.3: The Penrose diagram of a spacetime for gravitational collapse and black hole evaporation.
2.3.2 Alternative views on the Hawking radiation
Now the original derivation of the Hawking effect is presented, its relation to other physical mechanisms is given. The aim is to create a context for black hole radiation and to show how it perfectly connects with other ideas presented in this thesis.
2.3.2.1 Static observers and the Unruh effect
Return to the Schwarzschild solution
2GM 2GM −1 ds2 = 1 − dt2 − 1 − dr2 − r2dΩ2 , (2.193) r r and let ρ2 r − 2GM = . (2.194) 8GM Then 2GM (κρ)2 1 − = , (2.195) r 1 + (κρ)2 where κ = 1/4GM for the Schwarzschild black hole was used. In the region near the horizon, i.e. ρ 1, one finds 2GM 1 − ≈ (κρ)2 . (2.196) r Chapter 2. Quantum field theory in curved spacetime 95
From (2.194) one also has ρ dr = dρ . (2.197) 4GM And therefore dr2 = (κρ)2dρ2 . (2.198)
So in a small region outside the horizon, (2.193) can be written as
1 ds2 ≈ (κρ)2dt2 − dρ2 + dΩ2 , (2.199) 4κ2 where the last term represents a 2-sphere of radius 1/2κ. The first two terms can be rewritten as ds02 = ρ2d(κt)2 − dρ2 , (2.200) which after a comparison with (2.88) appears to be nothing but two-dimensional Rindler space. More specifically, region I of the maximally extended Schwarzschild spacetime (see section 1.5 of chapter 1) can be identified with the right Rindler wedge. So in this near-horizon Rindler description, the black hole horizon is an acceleration horizon. From the discussion of section 2.2 about the Unruh effect, one could therefore suspect that an observer on an orbit of ∂/∂(κt), i.e. a static observer just outside the horizon, would detect a thermal bath of particles. So the Unruh effect and the Hawking effect are perfectly consistent with each other. Of course, one should not take this analogy too literally since the Unruh effect takes place in flat Minkowski spacetime and the Hawking effect in a curved black hole spacetime. Nevertheless, the same physical principle seems to be at work in both cases.
Altough the proper acceleration of an ρ = constant worldline diverges as ρ → 0, its acceleration as measured by another ρ = constant observer will remain finite. Since
dτ 2 = ρ2d(κt)2 , (2.201) with ρ = a−1 constant, the acceleration as measured by an observer whose proper time is t is
dτ 1 1 = (κρ) = κ . (2.202) dt ρ ρ
But in Schwarzschild spacetime, an observer with proper time t is one at spatial infinity. This points out the equivalency between the Unruh temperature and the Hawking temperature and confirms the physical interpretation of the surface gravity given in section 1.6.
2.3.2.2 Heuristic arguments
To end the discussion on the origin of the Hawking radiation, two heuristic arguments are given which make the effect blend in with other physical phenomena.
First, recall that in section 1.4.1 it was mentioned that there exists an analytically solved model describing gravitational collapse of a spherically symmetric uniform dust cloud to a black hole. The solution existed of a matching of the Friedman-Lemaˆıtresolution on the interior of the cloud to the Schwarzschild solution on the outside. In the derivation of the Hawking radiation Chapter 2. Quantum field theory in curved spacetime 96 above it also became clear that the structure of spacetime just prior to the horizon formation is of crucial importance for the existence of the Hawking particles. And finally, in section 2.1.2 it was shown that there is particle creation in an expanding or contracting spacetime. It is now clear that these three ideas, presented in different contexts throughout this thesis, are perfectly consistent with the idea of Hawking radiation. So they present another viewpoint on the cre- ation of Hawking particles.
Another viewpoint that was already presented in Hawking’s orignal paper is that of negative energy flux across the horizon. One might picture this as follows. Just oustide the event horizon there will be virtual pairs of particles, one with negative energy and one with positive energy. The negative particle is in a regio which is classically forbidden but it can tunnel through the event horizon to the interior region. As seen chapter 1, the Killing vector field k representing time translations at infinity is space-like in this region. So the particle can exist as a real particle with a timelike momentum vector even thought its energy relative to infinity as measured by the Killing vector field k is negative. The other particle of the pair, having a positive energy, can escape to infinity where it consitutes a part of the Hawking radiation. The probability of the neg- ative energy particle tunnelling through the horizon is governed by the surface gravity since this quantity measures the gradient of the magnitude of the Killing vector. Or in other words, how fast the Killing vector is becomming spacelike. Instead of thinking of negative energy particles tunnelling through the horizon in the positive sense of time, one could regard them as positive energy particles crossing the horzon on past-directed world-lines and then being scattered onto future-directed world-lines by the gravitational field. However, it should be emphasized that this interpretation should not be taken to literally, certainly when recalling the problems of the particle interpretation of quantum field theory in curved spacetime as explained in section 2.1.3.
A final viewpoint is that a black hole is an excited state of the gravitational field which decays quantum mechanically and energy should be able to tunnel out of its potential well because of quantum fluctuations of the metric.
2.3.3 Trans-Planckian physics in Hawking radiation
After Hawking published his paper deriving the thermal spectrum of the radiation created by a black hole [57, 58], questions were raised about the use of paths from I− to I+. The frequencies of massless particles receive arbitrarily large redshifts along such paths as they pass through the collapsing dust cloud just prior to formation of the event horizon. The the range of frequencies that can be seen by distant observers at late times would have had to originate at I− with ultrahigh frequencies, including frequencies above the Planck scale. Local Lorentz invariance would be violated if such frequencies would be arbitrarily cut off. So the question was if the Hawking thermal spectrum would nevertheless survive the breaking of local Lorentz invariance. There is no conclusive answer to this question, but in the remaining of this section some models are presented that strongly hint that the physics at the Planck scale does not influence the Hawking spectrum.
In the context of black holes, Unruh [61] considered a definite model of sound waves propa- gating in a moving fluid that simulates the behavior of the event horizon of a black hole (see the Chapter 2. Quantum field theory in curved spacetime 97 water analogy in the introduction of chapter 1). By numerical methods he found that despite the breaking of Lorentz invariance in his fluid model, the sonic black hole nevertheless produced a spectrum of sound waves that was very close to a thermal spectrum. He demonstrated that the ultrahigh frequencies are not responsible for the thermal spectrum produced by a sonic black hole. This supports the viewpoint that the ultra high frequencies that appear in the derivation of the Hawking thermal spectrum in black hole evaporation are not necessarily essential for obtaining the thermal spectrum. In this context, related models with dispersion relations that break Lorentz invariance have been considered by, for example, Jacobson [62].
In [63] the Hadamard form of the two-point correlation function of the field at very short distances characterized by an invariant Planck length was altered. The invariance of the Planck length appearing in the two-point function is enforced by means of a non-linear physical real- ization of the Lorentz group. It was shown that this alteration of the Hadamard form at the invariant Planck scale has negligible effect on the thermal spectrum of Hawking radiation. This conclusion extends to spectral frequencies much higher than the energy scale set by the Hawking temperature of the black hole. Thus, the thermal spectrum of an evaporating black hole of radius above the Planck scale appears to be insensitive to such changes in physics near the Planck scale.
In Deser and Levin [64, 65], the spacetime of a four-dimensional black hole is embedded in a six-dimensional Minkowski spacetime in a global way, in the sense that the embedding in the six- dimensional flat spacetime covers the usual Kruskal maximal extrension of Schwarzschild space- time (with a white hole in it) without encountering a coordinate singularity at the Schwarzschild radius of the black hole. In this embedding, a detector held at rest at constant Schwarzschild radial distance r is mapped to a detector moving at constant acceleration in the six-dimensional Minkowski spacetime. It is shown that the temperature a/2π of the thermal spectrum measured by this uniformly accelerated detector is the temperature that the detector would detect as a result of the Hawking radiation. This correspondence makes no use of trans-Planckian frequen- cies and thus supports the view that they are not essential to the thermal spectrum of Hawking radiation.
The string theory derivation of the Hawking radiation for a nearly extremal supersymmetric black hole [66] makes use of the Minkowski spacetime limit of the black hole in terms of D-branes and oppositely moving string excitations that interact and produce the Hawking thermal spec- trum of radiation, including the gray-body factor, without appealing to large red- or blueshifts. This again suggests that the thermal spectrum is not dependent on very high frequency modes of the radiation field.
2.4 Angular momentum and gray body factors
In this section the role of the gray body factors of the black hole spectrum Γlm(ω), that were encountered in the derivation of the Hawking radiation, will be discussed. In particular, the focus will be on their relation with the angular momentum of the particles. The derivation is based upon [9]. Chapter 2. Quantum field theory in curved spacetime 98
Again, a massless scalar field ψ is considered in a Schwarzschild background. It is of advantage in this section to use the tortoise coordinates as introduced in chapter 1. With the metric in tortoise coordinates (1.52), the action for ψ can be written as
1 Z S = d4x (−g)1/2gµν∂ ψ∂ ψ (2.203) 2 µ ν " 2 2 2 2# 1 Z (∂ ψ) − (∂ ∗ ψ) 1 ∂ψ 1 ∂ψ = dt dr∗ dθ dφ t r − − F r2 sin θ 2 F r2 ∂θ r2 sin2 θ ∂φ